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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

Logarithmic: logax = y

Exponential: ay = x

2

3

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

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

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.

Article How to Understand Logarithms provided by wikiHow.
Content on wikiHow can be shared under a Creative Commons License.

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