How to Calcolate Capacitive Reactance

Capacitive Reactance

Capacitive reactance is how the impedance (or resistance) of a capacitor changes in regard to the frequency of the signal passing through it.

Capacitors, unlike resistors, are reactive devices. This means that they offer different resistances to signals of differing frequencies. To low-frequency signals, capacitors offer a lot of resistance, so that low-frequency signals are essentially blocked from passing through a capacitor. As the frequencies of the signals passing through a capacitor increase, the capacitor offers less and less resistance, so that higher frequency signals can pass through easier.

The formula for calculating capacitive reactance is:

XC is equal to the value of the capacitor reactance or impedance.

You can see, according to this formula, how capacitive reactance changes according to the frequency of the signal input into it. It has an indirect relationship with the frequency of the signal passing through it. The higher the frequency of the signal, the lower the value of the reactance. The lower the frequency of the signal, the higher the value of the reactance. 

What is a Bypass Capacitor?

A bypass capacitor is a capacitor that shorts AC signals to ground, so that any AC noise that may be present on a DC signal is removed, producing a much cleaner and pure DC signal.

A bypass capacitor essentially bypasses AC noise that may be on a DC signal, filtering out the AC, so that a clean, pure DC signal goes through without any AC ripple.

For example, you may want a pure DC signal from a power source.

Below is a transistor circuit. A transistor is an active device, so in order to work, it needs DC power. This power source is VCC. In this case, VCC equals 15 volts.

This 15 volts provides power to the transistor so that the transistor can amplify signals. We want this signal to be as purely DC as possible. Although we obtain our DC voltage, VCC, from a DC power source such as a power supply, the voltage isn't always purely DC. In fact, many times the voltage is very noisy and contains a lot of AC ripple on it, especially at the 60Hz frequency because this is the frequency at which AC signals run in many countries.

So although we want a pure DC signal, such as below:

 

Many times, we get a noisy signal that looks like:

A DC signal such as this is actually very common. This is undesired because it adds noise to the transistor circuit. Therefore, this noisy DC signal will be imposed on the AC signal. So the AC signal which may have music or some type of recording will now have much more noise.

This noise which is on the signal is AC ripple. Many times when using a DC power supply connected to an AC power outlet, it will have some of the AC noise transfer to the DC power voltage. AC ripple can also appear from other sources, so even batteries can produce noise.

To eliminate this AC ripple, we use a bypass capacitor. So our transistor circuit above will have a bypass capacitor added to it:

A capacitor is a device that offers a tremendously high resistance for signals of low frequencies. Therefore, signals at low frequencies will not go through them. This is because signals (current) always takes the path of least resistance. Therefore, they will instead go through the resistor, RE. Remember, again, this is for low frequency signals, which is basically DC signals.

However, capacitors offer much less resistance at higher frequencies (AC signals). So AC signals will go through the capacitor and then to gorund. Therefore, DC signals will go through the resistor, RE, while AC signals will go through the capacitor, getting shunted to ground. So AC signals get shunted to ground. This is how we have a clean DC signal across our circuit, while AC noise imposed on it is bypassed to ground.

 

So a bypass capacitor blocks the DC from entering it by the great resistance it offers to the signal but accepts the AC noise that may be on the DC line and shunts or bypasses it to ground. This is how bypass capacitors work. 

What is a Smoothing Capacitor?

A Smoothing capacitor is a capacitor that acts to smooth or even out fluctations in a signal.

The most common and used application for smoothing capacitors is after a power supply voltage or a rectifier.Power supply voltage can sometimes supply erratic and unsmooth voltages that fluctuate greatly.When a steady DC signal is needed and is necessary, a smoothing capacitor is the right component needed in order to smooth out the fluctuating signal to make it more steady.

We'll go over an example of this now.

A prime example of when a smoothing capacitor is used is in conjunction with a rectifier circuit.

If you place a resistor in series with a diode and then input an AC signal into the circuit:

Now if you place a smoothing capacitor in parallel with the diode, like this, the resulting waveform will be:

You can see now how much smoother the waveform is. It no longer goes all the way down to zero and back up. The capacitor charges up from 0 to the top of the waveform and then discharges from 0 to the bottom of the waveform. This charging and discharging smooths out the waveform so that it doesn't hit the extreme ups and downs. Thus, a smoothing capacitor is extremely useful in cases of fluctuating signals that need to be more constant and steady.

Usually when choosing a smoothing capacitor, an electrolytic capacitor is used from anywhere from 10µF to a few thousand µF. The greater the amplitude of the fluctations and the greater the waveform, the larger capacitor will be necessary. Thus, if you're smoothing a 30mV waveform, a 10µF capacitor may suffice to smooth out the signal. However, if you're dealing with a much greater signal, you will need a much larger capacitor, say, maybe 3300µF in order to smooth it out to a near DC level. Experiment with the capacitors. Check the signal on an oscilloscope to see which capacitor suffices best and is best for the circuit at hand. 

What is a Coupling Capacitor?

Use of Coupling CapacitorsA coupling capacitor is a capacitor which is used to couple or link together only the AC signal from one circuit element to another. The capacitor blocks the DC signal from entering the second element and, thus, only passes the AC signal.

Coupling capacitors are useful in many types of circuits where AC signals are the desired signals to be output while DC signals are just used for providing power to certain components in the circuit but should not appear in the output.

For example, a coupling capacitor normally is used in an audio circuits, such as a microphone circuit. DC power is used to give power to parts of the circuit, such as the microphone, which needs DC power to operate. So DC signals must be present in the circuit for powering purposes. However, when a user talks into the microphone, the speech is an AC signal, and this AC signal is the only signal in the end we want passed out. When we pass the AC signals from the microphone onto the output device, say, speakers to be played or a computer to be recorded, we don't want to pass the DC signal; remember, the DC signal was only to power parts of the circuit. We don't want it showing up on the output recording. On the output, we only want the AC speech signal. So to make sure only the AC passes while the DC signal is blocked, we place a coupling capacitor in the circuit.

How to Place a Coupling Capacitor in a Cirucit

In order to place a capacitor in a circuit for AC coupling, the capacitor is connected in series with the load to be coupled.

A capacitor is able to block low frequencies, such as DC, and pass high frequencies, such as AC, because it is a reactive device. It responds to different frequencies in different ways. To low frequency signals, it has a very high impedance, or resistance, so low frequency signals are blocked from going through. To high frequency signals, it has a low impedance or resistance, so high frequency signals are passed through easily.

How To Test a Capacitor

In this article, we will go over different tests that we can use to tell whether a capacitor is good or not, all by utilizing the functions of a digital multimeter.

There are many checks we can do to see if a capacitor is functioning the way it should. We will use and exploit the characteristics and behaviors that a capacitor should show if it is good and, in thus doing so, determine whether its is good or defective.

So let's start:

Test a Capacitor with an Ohmmeter of a Multimeter

A very good test you can do is to check a capacitor with your multimeter set on the ohmmeter setting.

By taking the capacitor's resistance, we can determine whether the capacitor is good or bad.

To do this test, We take the ohmmeter and place the probes across the leads of the capacitor. The orientation doesn't matter, because resistance isn't polarized.

If we read a very low resistance (near 0Ω) across the capacitor, we know the capacitor is defective. It is reading as if there is a short across it.

If we read a very high resistance across the capacitor (several MΩ), this is a sign that the capacitor likely is defective as well. It is reading as if there is an open across the capacitor.

A normal capacitor would have a resistance reading up somewhere in between these 2 extremes, say, anywhere in the tens of thousands or hundreds of thousand of ohms. But not 0Ω or several MΩ.

This is a simple but effective method for finding out if a capacitor is defective or not.

Test a Capacitor with a Multimeter in the Capacitance Setting

Another check you can do is check the capacitance of the capacitor with a multimeter, if you have a capacitance meter on your multimeter. All you have to do is read the capacitance that is on the exterior of the capacitor and take the multimeter probes and place them on the leads of the capacitor. Polarity doesn't matter.

This is the same as the how the setup is for the first illustration, only now the multimeter is set to the capacitance setting.

You should read a value near the capacitance rating of the capacitor. Due to tolerance and the fact that (specifically, electrolytic capacitors) may dry up, you may read a little less in value than the capacitance of the rating. This is fine. If it is a little lower, it is still a good capacitor. However, if you read a significantly lower capacitance or none at all, this is a sure sign that the capacitor is defective and needs to be replaced.

Checking the capacitance of a capacitor is a great test for determining whether a capacitor is good or not.

Test a Capacitor with a Voltmeter

Another test you can do to check if a capacitor is good or not is a voltage test.

A test that you can do is to see if a capacitor is working as normal is to charge it up with a voltage and then read the voltage across the terminals. If it reads the voltage that you charged it to, then the capacitor is doing its job and can retain voltage across its terminals. If it is not charging up and reading voltage, this is a sign the capacitor is efective. Afterall, capacitors are storage devices. They store a potential difference of charges across their plate, which are voltages. The anode has a positive voltage and the cathode has a negative voltage.

To charge the capacitor with voltage, apply DC voltage to the capacitor leads. Now polarity is very important for polarized capacitors (electrolytic capacitors). If you are dealing with a polarized capacitor, then you must observe polarity and the correct lead assignments. Positive voltage goes to the anode (the longer lead) of the capacitor and negative or ground goes to the cathode (the shorter lead) of the capacitor. Apply a voltage which is less than the voltage rating of the capacitor for a few seconds. For example, feed a 25V capacitor 9 volts and let the 9 volts charge it up for a few seconds. As long as you're not using a huge, huge capacitor, then it will charge in a very short period of time, just a few seconds. After the charge is finished, disconnect the capacitor from the voltage source and read its voltage with the multimeter. The voltage at first should read near the 9 volts (or whatever voltage) you fed it. Note that the voltage will discharge rapidly and head down to 0V because the capacitor is discharging its voltage through the multimeter. However, you should read the charged voltage value at first before it rapidly declines. This is the behavior of a healthy and a good capacitor. If it will not retain voltage, it is defective and should be replaced.

So there you have it, 4 strong tests that you can do (all or either/or) to test whether a capacitor is good or not. 

What are Capacitors Used For?

Capacitors, either standalone or used with other electronic components such as resistors and inductors, have a wide variety of uses in circuits. What capacitors are used for are shown below:

1) RC Timing Circuit

A capacitor, when combined with a resistor, is used to form a RC circuit, which acts as a timing mechanism. The combination of the value of the resistance of the resistor and the value of the capacitance of the capacitor determines how long it takes the capacitor to charge up or to discharge in a circuit. By using the needed values, a precise timing sequence can be made that is needed for a circuit.

The product of the RC value is called the time constant. It is this RC product, which is measured in unit seconds, which decides the timing interval of the RC circuit.

2) AC Coupling

A capacitor is used for AC coupling. This is where a capacitor couples, or transfers, the AC portion of a signal to output and blocks the DC from being transmitted. This is necessary in situations where the AC aspect of a signal needs to be passed as output but not the DC. In this way, the coupling capacitor acts as a type of filter, passing AC and blocking DC.

A prime example of where coupling capacitors are needed is in microphone circuits. Microphones need DC power in order to operate, in order to turn on. However, this DC output should not appear in the output; it's only for powering the microphone. The only output we want is the user's speech, music, etc, which are AC signals. If the DC were placed in output, it could cause DC offset, which means the signal could shift up or down. To counter this, coupling capacitors remove all aspects of DC and only pass the wanted AC (music, speech, etc.)

To find out more about coupling capacitors, visit the link coupling capacitors for more in-detail information.

3) Removes AC Noise From DC Signals

A capacitor can also be used as a bypass capacitor, which is a capacitor that shunts AC signals of a DC to ground. This cleans any AC noise that may be on DC signals, allowing a much cleaner DC signal to be used in a circuit.

As you can see above, the capacitor in parallel to the resistor, RE serves as a bypass capacitor. This bypass capacitor bypasses the AC aspect of the signal to ground, allowing a pure DC signal to go through the resistor, RE. This helps the transistor eliminate any noise that may enter it and allows it to produce a cleaner output signal.

To find out more about bypass capacitors, visit the link, bypass capacitors for more in-detail inofrmation.

4) Power Supply Filter

Capacitors are used to smooth the pulsating voltage from a power supply into a steady direct current (DC).

If a typical sine wave current needs to be converted into a steady DC current, the above circuit allows it to do so. A rectifier rectifies the AC signal so that it remains above the positive threshold, producing a pulsating DC signal. A capacitor then placed in parallel to the output smoothes the signal so that the DC output is steady in value.

5) Spike Remover

Digital logic circuits can use lots of momentary current when they switch from off to on and vice versa. This can cause very brief but substantial reductions in power applied to nearby circuits. These power spikes (or glitches, as they are called sometimes) can be eliminated by placing a small (0.1µF) capacitor across the power leads of the logic circuit:

A capacitor acts like a miniature battery that supplies power during the spike.

6) RC Integrator

A capacitor can also be used as an RC integrator, which is a circuit which can act as a low pass filter. A basic RC integrator is shown below:

If the input pulses are speeded up, the output waveforms (often called sawtooth) will not reach their full amplitude. Therefore, this RC integrator will reproduce signals in full which have an frequency below a certain level but not those above a certain frequency. It acts as a low-pass filter. This is good for many applications where only low-frequency signals need to be output but not higher frequency signals.

7) RC Differentiator

A capacitor can also be used as an RC differentiator. Below is a basic example of one:

An RC Differeniating circuit produces symmetrical output waves with sharp positive and negative peaks. It's used ot make narrow pulse generators for television receivers and to trigger digital logic circuits. 

How to Create a Non-Polar Capacitor From a Polar Type ???

Instructions.....

1. • 1
     Examine the capacitors. The body has a polarity mark on one side, such as a minus sign. One lead may be shorter than the other; if you see this, the shorter lead is the negative side.
   • 2
     Solder the negative leads of the two capacitors together, leaving the positive leads free.
   • 3
     Turn the multimeter's function knob to the capacitance setting. Touch the tips of the meter's probes to the capacitor's leads. Each lead should be connected to one probe tip; do not short the leads together. Observe a reading of about 50 microfarads on the meter's display. Two capacitors having the same value, connected in series, result in a total capacitance of half the original value.

Just Do a Timer Circuit with 4060B

Timer Circuits With 4060B

There are many applications for which a timer is very useful to turn a device on or off automatically after a preset interval - for example, switching off an irrigation system after 30 minutes of use, turning off a battery charger to prevent overcharging, etc.

Timing short intervals of milliseconds to minutes can easily be achieved using a NE555 timer chip. Unfortunately, this device is not suitable for timing longer intervals, and so a suitable alternative is required.

Binary Counting with the 4060B

The 4060B (pictured above) is a CMOS binary counter. Using a resistor and a capacitor, the counting speed can be set by the user very easily. The pins of the 4060B integrated circuit output the running count in binary as shown below:

0 = 0000000000
1 = 0000000001
2 = 0000000010
3 = 0000000011
4 = 0000000100
5 = 0000000101
6 = 0000000110
7 = 0000000111
8 = 0000001000

Each of the binary 1's and 0's is called a bit (much as the numbers 0,1,2...8,9 are called digits in the decimal number system). The furthest right bit represents 1, the next to the left represents 2, the next represents 4, the next 8, the next 16 and so on doubling every time you move one position to the left. Therefore 000010000 is binary for 16, and 000100000 is binary for 32.

To keep things simple, let's assume the count is increased by one every second. The rightmost bit(the 1's bit) will be off for one second, on for one second, off for one second and so on...

0000000001, 0000000010, 0000000011

The fifth bit from the right (the 16's bit) is therefore off for 16 seconds (when the count is 0-15), then on for 16 seconds (when the count is 16-31), then off for 16 seconds (when the count is 32-47), and so on.

With this knowledge, we can make a very accurate timer utilising our 4060B binary counter chip. Let's say we want a 16 second timer: we start the 4060B counter from 0, and wait until the 16's bit goes from 0 to 1. At that exact time we know that 16 seconds have elapsed. Similarly if we start the counter again, and wait until the 32's bit goes from 0 to 1, we know that 32 seconds have elapsed.

A timer which can only time, 1, 2, 4, 8, 16, 32, 64, 128, and so on seconds would not be very useful, but since we can adjust the speed of the count, any time interval from seconds to 24+ hours can be accurately timed.

4060B Timer

A schematic of the 4060B chip is provided below:

The pins labelled in red Q4-Q14 are the binary outputs: Q4 for the 16's, Q5 for the 32's, Q6 for the 64's and so on up to Q13 for the 8192's, and Q14 for the 16384's.

Just three external components are required to control the 4060B counter - two resistors and one capactor. The frequency of the internal oscillator (i.e. the speed of the count) is set according to the equation given at the bottom of the schematic below:

Since Q14 represents the 16,384's and Q4 represents the 16's - we know it will take 1,024 times longer (16,384 / 16) for Q14 to flip from 0 to 1 than it takes Q4. So, for an example 2-hour timer (=7,200 seconds), we just need to fine-tune the circuit so that Q4 turns on after 7,200 / 1,024 seconds = 7.03 seconds, knowing that if that is done correctly, after exactly 2 hours Q14 will flip from 0 to 1.

Putting Together the Timer Circuit

The circuit shown above  is a timer which energises a relay after a preset time has elapsed. It can be set to time an interval from 30 seconds to 24 hours.

The orange arrow labelled Range should be connected to a pin on the 4060B chip selected from theRANGE table. If for example, you require a timer to time 3 hours, connect it to pin number 1 on the chip since that pin corresponds to the time range 2hrs to 4hrs. 

3 hours is 10,800 seconds, and we are using the output from pin 1 to trigger the relay. Looking at theSETUP table entry for pin 1 we see that we divide our target time (10,800 seconds) by 256 to obtain the on/off time for the yellow LED at pin 7 = 42.28 seconds. Therefore, if we adjust the potentiometerR4 so that the yellow LED turns on after approximately 42 seconds, we'll know that the relay will be energised after approximately 3 hours.

The diode D1 makes this a one-shot timer. This means that after the programmed time delay of 3 hours, the relay will stay on until the circuit is reset. If the diode is omitted from the circuit then you get a repeating timer with the relay off for 3 hours, on for 3 hours, off for 3 hours, and so on until the circuit it reset.

What is e ???...Must read it.... :-) ...

e (Euler's Number)

The number e is a famous irrational number, and is one of the most important numbers in mathematics.

The first few digits are:

2.7182818284590452353602874713527 (and more ...)

It is often called Euler's number after Leonhard Euler

e is the base of the Natural Logarithms (invented by John Napier).

e is found in many interesting areas, so it is worth learning about.

Calculating

There are many ways of calculating the value of e, but none of them ever give an exact answer, because e is irrational (not the ratio of two integers).

But it is known to over 1 trillion digits of accuracy!

For example, the value of (1 + 1/n)ⁿ approaches e as n gets bigger and bigger:

n (1 + 1/n)n 1 2.00000 2 2.25000 5 2.48832 10 2.59374 100 2.70481 1,000 2.71692 10,000 2.71815 100,000 2.71827

Another Calculation

The value of e is also equal to 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ... (etc)

(Note: "!" means factorial)

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.718055556

And you can try that yourself at Sigma Calculator.

Remembering

To remember the value of e (to 10 places) just remember this saying (count the letters!):

• To
• express
• e
• remember
• to
• memorize
• a
• sentence
• to
• simplify
• this

Or you can remember the curious pattern that after the "2.7" the number "1828" appears TWICE:

2.7 1828 1828

And following THAT are the angles 45°, 90°, 45° in a Right-Angled Isosceles (two equal angles) Triangle:

2.7 1828 1828 45 90 45

(An instant way to seem really smart!)

An Interesting Property

Just for fun, try "Cut Up Then Multiply"

Let us say that we cut a number into equal parts and then multiply those parts together.

Example: Cut 20 into 4 pieces and multiply them:

Each "piece" is 20/4 = 5 in size

5×5×5×5 = 5⁴ = 625

Now, ... how could we get the answer to be as big as possible, what size should each piece be?

Example continued: try 5 pieces

Each "piece" is 20/5 = 4 in size

4×4×4×4×4 = 4⁵ = 1024

Yes, the answer is bigger! But is there a best size?

The answer: make the parts "e" (or as close to e as possible) in size.

Example: 10

10 cut into 3 parts is 3.3...	3.3...×3.3...×3.3... (3.3...)3 = 37.037...
10 cut into 4 equal parts is 2.5	2.5×2.5×2.5×2.5 = 2.54 = 39.0625
10 cut into 5 equal parts is 2	2×2×2×2×2 = 25 = 32

The winner is the number closest to "e", in this case 2.5.

Try it with another number yourself, say 100, ... what do you get?

Advanced: Use of e in Compound Interest

Often the number e appears in unexpected places.

For example, e is used in Continuous Compounding (for loans and investments):

Formula for Continuous Compounding

Why does that happen?

Well, the formula for Periodic Compounding is:

FV = PV (1+r/n)ⁿ

where FV = Future Value
PV = Present Value
r = annual interest rate (as a decimal)
n = number of periods

But what happens when the number of periods heads to infinity?

The answer lies in the similarity between:

(1+r/n)n	and	(1 + 1/n)n
Compounding Formula		e (as n approaches infinity)

By substituting x = n/r :

• r/n becomes 1/x and
• n becomes xr

And so:

(1+r/n)n

becomes

(1+(1/x))xr

Which is just like the formula for e (as n approaches infinity), with an extra r as an exponent.

So, as x goes to infinity, then (1+(1/x))^xr goes to e^r

And that is why e makes an appearance in interest calculations!

What is log ???...A crispy Note.....

Introduction to Logarithms

The Word

"Logarithm" is a word made up by Scottish mathematician John Napier (1550-1617), from the Greek word logos meaning "proportion, ratio or word" and arithmos meaning "number", ... which together makes "ratio-number" !

In its simplest form, a logarithm answers the question:

How many of one number do we multiply to get another number?

 

Example: How many 2s do we multiply to get 8?

Answer: 2 × 2 × 2 = 8, so we needed to multiply 3 of the 2s to get 8

So the logarithm is 3

 

How to Write it

We would write "the number of 2s you need to multiply to get 8 is 3" as

log₂(8) = 3

So these two things are the same:

The number we are multiplying is called the "base", so we would say:

• "the logarithm of 8 with base 2 is 3"
• or "log base 2 of 8 is 3"
• or "the base-2 log of 8 is 3"

Notice we are dealing with three numbers:

• the base: the number we are multiplying (a "2" in the example above)
• how many times to use it in a multiplication (3 times, which is the logarithm)
• The number we want to get (an "8")

More Examples

Example: What is log₅(625) ... ?

We are asking "how many 5s need to be multiplied together to get 625?"

5 × 5 × 5 × 5 = 625, so we need 4 of the 5s

Answer: log₅(625) = 4

Example: What is log₂(64) ... ?

We are asking "how many 2s need to be multiplied together to get 64?"

2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s

Answer: log₂(64) = 6

Exponents

Exponents and Logarithms are related, let's find out how ...

The exponent says how many times to use the number in a multiplication. In this example: 23 = 2 × 2 × 2 = 8 (2 is used 3 times in a multiplication to get 8)

So a logarithm answers a question like this:

In this way:

The logarithm tells you what the exponent is!

In that example the "base" is 2 and the "exponent" is 3:

So the logarithm answers the question:

What exponent do we need 
(for one number to become another number) ?

The general case is:

Example: What is log₁₀(100) ... ?

10² = 100

So an exponent of 2 is needed to make 10 into 100, and:

log₁₀(100) = 2

Example: What is log₃(81) ... ?

3⁴ = 81

So an exponent of 4 is needed to make 3 into 81, and:

log₃(81) = 4

Common Logarithms: Base 10

Sometimes you will see a logarithm written without a base, like this:

log(100)

This usually means that the base is really 10.

It is called a "common logarithm". Engineers love to use it. On a calculator it is the "log" button.

It is how many times you need to use 10 in a multiplication, to get the desired number.

Example: log(1000) = log₁₀(1000) = 3

Natural Logarithms: Base "e"

Another base that is often used is e (Euler's Number) which is approximately 2.71828.

This is called a "natural logarithm". Mathematicians use this one a lot. On a calculator it is the "ln" button.

It is how many times you need to use "e" in a multiplication, to get the desired number.

Example: ln(7.389) = log_e(7.389) ≈ 2

Because 2.71828² ≈ 7.389

But Sometimes There Is Confusion ... !

Mathematicians use "log" (instead of "ln") to mean the natural logarithm. This can lead to confusion:

Example	Engineer Thinks	Mathematician Thinks
log(50)	log10(50)	loge(50)	confusion
ln(50)	loge(50)	loge(50)	no confusion
log10(50)	log10(50)	log10(50)	no confusion

So, be careful when you read "log" that you know what base they mean!

Logarithms Can Have Decimals

All of our examples have had whole number logarithms (like 2 or 3), but logarithms can have decimal values like 2.5, or 6.081, etc.

Example: what is log₁₀(26) ... ?

Get your calculator, type in 26 and press log Answer is: 1.41497...

The logarithm is saying that 10^1.41497... = 26
(10 with an exponent of 1.41497... equals 26)

This is what it looks like on a graph: See how nice and smooth the line is.

Read Logarithms Can Have Decimals to find out more.

Negative Logarithms

-	Negative? But logarithms deal with multiplying. What could be the opposite of multiplying? Dividing!

A negative logarithm means how many times to divide by the number.

We could have just one divide:

Example: What is log₈(0.125) ... ?

Well, 1 ÷ 8 = 0.125, so log₈(0.125) = -1

Or many divides:

Example: What is log₅(0.008) ... ?

1 ÷ 5 ÷ 5 ÷ 5 = 5^-3, so log₅(0.008) = -3

It All Makes Sense

Multiplying and Dividing are all part of the same simple pattern.

Let us look at some Base-10 logarithms as an example:

	Number	How Many 10s	Base-10 Logarithm
	.. etc..
	1000	1 × 10 × 10 × 10	log10(1000)	=	3
	100	1 × 10 × 10	log10(100)	=	2
	10	1 × 10	log10(10)	=	1
	1	1	log10(1)	=	0
	0.1	1 ÷ 10	log10(0.1)	=	-1
	0.01	1 ÷ 10 ÷ 10	log10(0.01)	=	-2
	0.001	1 ÷ 10 ÷ 10 ÷ 10	log10(0.001)	=	-3
	.. etc..

If you look at that table, you will see that positive, zero or negative logarithms are really part of the same (fairly simple) pattern.