# Basic Description

Logarithms are considered the inverse or opposite operations to exponents, just as subtraction is the inverse to to addition or square rooting is the inverse to squaring.

For example, suppose we have the exponential expression , which we know will equal 8. Now, suppose we want to do the inverse operation and go from the value 8 to the exponent 3 with a base of 2. We could do the inverse operation by using logarithms and write , which is read "logarithm of 8 base 2 is equal to 3".

- In order words, if we have an exponential equation:

- we can write an equivalent logarithmic equation:

What about exponential equations in the form ? It might seem harder to solve for x in this case because there is no whole number exponent that will give us the value of 932 with a base of 10. However, if we simply rewrite the equation as an logarithmic equation , we can find quite easily with a calculator that x is about 2.969.

To look at some more examples of switching between exponential and logarithmic equations:
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# A More Mathematical Description

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**Definition of a Logarithm**

if and only if , where b > 1 and x > 0

In words: The logarithm of a value at a given **base** is the **power** (exponent) that the base must be raised to produce the value.

## Bases

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As seen from the definition above, the base of a logarithm affects how a logarithm is evaluated. Bases can be any positive number except for 1, and the logarithms of a value can be found at different bases using a change of base formula.

### Common Bases

There are three main bases that are most frequently used:

### Changing Bases

To go from a logarithm of base *k* to a logarithm of base *a*, we use the formula:

## Graphing a Logarithmic Function

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Graph of a logarithmic function

Graph of a three common logarithmic functions

The image on the left is a graph of a basic logarithmic function: . A logarithmic function, such as the one used to create the featured image, takes the basic form , where *b* is fixed while *y* and *x* are variables.

As you can see from the graph, there is an vertical asymptote An asymptote is a curve (in this case a vertical line) that a function comes infinitely closer to, but never touches or intersects. Thus, for the function , at , at x = 0, so that logarithmic functions are undefined when x is less than or equal to 0. However, non-real logarithms for negative x values can be found using complex logarithms with complex numbers.

The second image compares the graphical representations of the most common logarithmic functions:

,

, and

.

## Basic Properties of Logarithms

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