Simple step-by-step instructions

# How to Understand Logarithms 1234 views, 6 comments

Confused by logs? Don't worry! A logarithm (log for short) is actually just an exponent in a different form. logax = y is the same as ay = x.[1]
Instructions
1
Know the difference between logarithmic and exponential equations. This is very simple. If it contains a logarithm it is logarithmic. A logarithm is denoted by the letters "log". If the equation contains an exponent it is an exponential equation. An exponent is a superscript number placed after a number.

Logarithmic: logax = y

Exponential: ay = x

2
Know the parts of a logarithm. The base is the subscript number found after the letters "log"--2 in this example. The argument or number is the number following the subscript number--8 in this example. Lastly, the answer is the number that the logarithmic expression is set equal to--3 in this equation.[2]

3
Know the difference between a common log and a natural log.

Common logs have a base of 10. (ex. log10x). If a log is written without a base (as log x), then it is assumed to have a base of 10.

Natural logs: These are logs with a base of e. e is a mathematical constant that is equal to the limit of (1 + 1/n)n as n approaches infinity, approximately 2.718281828. (It has many more digits than those written here.) logex is often written as ln x.

4
Know and apply the properties of logarithms. The properties of logarithms allow you to solve logarithmic and exponential equations that would be otherwise impossible. These only work if the base a and the argument are positive. Also the base a cannot be 1 or 0. The properties of logarithms are listed below with a separate example for each one with numbers instead of variables. These properties are for use when solving equations.

loga(xy) = logax + logay
A log of two numbers, x and y, that are being multiplied by each other can be split into two separate logs: a log of each of the factors being added together. (This also works in reverse.)

Example:
log216 =
log28*2 =
log28 + log22

loga(x/y) = logax - logay
A log of a two numbers being divided by each other, x and y, can be split into two logs: the log of the dividend x minus the log of the divisor y.

Example:
log2(5/3) =
log25 - log23

loga(xr) = r*logax
If the argument x of the log has an exponent r, the exponent can be moved to the front of the logarithm.

Example:
log2(65)
5*log26

loga(1/x) = -logax
Think about the argument. (1/x) is equal to x-1. Basically this is another version of the previous property.

Example:
log2(1/3) = -log23

logaa = 1
If the base a equals the argument a the answer is 1. This is very easy to remember if one thinks about the logarithm in exponential form. How many times should one multiply a by itself to get a? Once.

Example:
log22 = 1

loga1 = 0
If the argument is one the answer is always zero. This property holds true because any number with an exponent of zero is equal to one.

Example:
log31 =0

(logbx/logba) = logax
This is known as "Change of Base".[3] One log divided by another, both with the same base b, is equal to a single log. The argument a of the denominator becomes the new base, and the argument x of the numerator becomes the new argument. This is easy to remember if you think about the base as the bottom of an object and the denominator as the bottom of a fraction.

Example:
log25 = (log 5/log 2)

5
Practice using the properties. These properties are best memorized by repeated use when solving equations. Here's an example of an equation that is best solved with one of the properties:

4x*log2 = log8 Divide both sides by log2.
4x = (log8/log2) Use Change of Base.
4x = log28 Compute the value of the log.
4x = 3 Divide both sides by 4.
x = 3/4 Solved.

Tips and Warnings:

• "2.7jacksonjackson" is a useful mnemonic device for e. 1828 is the year Andrew Jackson was elected, so the mnemonic stands for 2.718281828.
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What are your thoughts (6 Responses)

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Peter L. Griffiths
posted on Apr 2, 2015 11:18:38 am
In paragraph 44 of the Constructio, Napier rather obscurely states a formula for arriving at the logarithms of the sines of angles between 75 degrees and 90 degrees, this is 10^7 X (0.9999999)^(5000s)equals 10^7 -(5000s) + (5/4)s^2. s should not exceed 346574 divided by 5000.

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Peter L.Griffiths
posted on Feb 13, 2014 02:43:28 pm
Further to my comments in January 2014, Napier and before him Regiomontanus were able to construct sine and cosine tables by applying the formula sin2u= 2sinu.cosu =2s(1-s^2)^0.5. Also sin30 is initially known to be 1/2, but by applying this formula and quadratic equations, sin15 can be obtained. Sin15 is the same as cos75, so sin75 can be obtained, also its power relationship to both sin30 and sin45 can be seen. This could be of considerable help in constructing log tables.

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Peter L. Griffiths
posted on Jan 17, 2015 11:48:11 am
Further to my previous comments, one of the great mysteries of the discovery of logarithms is why logarithms to base 2 do not seem to have been published both for their own sake as well as for a check on logarithms to other bases.

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Peter L. Griffiths
posted on Jan 29, 2014 11:13:00 am
Logarithms discovered by Napier in 1614 were based on sine tables with 0.9999999 just below sine 90 degrees as the base which is raised to successive powers. Initially the results are nearly equal to the shortfall from 1.0000000. It would be a very onerous task to raise these powers from sine 90 degrees down to sine 1 degree, but this would be helped by sine 75 degrees equalling 0.9659258 being raised to the power of 10 and equalling sine 45 degrees which is 0.7070299.

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Peter L. Griffiths
posted on Jan 17, 2014 11:45:00 am
Some of my dates are wrong, 1768 should be 1668, and 1767 should be 1676, apologies.

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Peter L. Griffiths
posted on Jan 17, 2014 11:39:59 am
Logarithms were discovered by Napier in 1614, probably the most important British contribution to maths. Most modern mathematicians have no idea how Napier did it, but the table of sines was of considerable assistance. Further developments took place in 1768 when Nicholas Mercator recognised that the area under a symmetrical ellipse measured the logarithm of the distance along the x axis. Isaac Newton in 1767 reversed the integrated elliptical formula to produce the infinite series for the antilogarithm e which would not otherwise be known.