# Algebra/Logarithms

Logarithms (commonly called "logs") are a specific instance of a function being used for everyday use. Logarithms are used commonly to measure earthquakes, distances of stars, economics, and throughout the scientific world.

## Contents

## Logarithms[edit]

In order to understand logs, we need to review exponential equations. Answer the following problems:

1) What is 4 to the power of 3?

2) What is 3 to the power of 4?

3)

4)

After you finish, check your answers:

1: "4 to the power of 3" means "4 multiplied 3 times." Thus, 4 to the power of 3 equals 4 times 4 times 4. 4 times 4=16. 16 times 4=64. So 4 to the power of 3 equals 64.

2: 3 times 3 = 9. Times 3 equals 27. Times 3 equals 81. So 3 to the power of 4 equals 81.

3:

4:

Just like there is a way to say and write "4 to the power of 3" or ", there is a specific way to say and write logarithms.

For example, "4 to the power of 3 equals 64" can be written as:

However, it can also be written as:

Once, you remember that the base of the exponent is the number being raised to a power and that the base of the logarithm is the subscript after the log, the rest falls into place. I like to draw an arrow (either mentally or physically) from the base, to the exponent, to the product when changing from logarithmic form to exponential form. So visually or mentally I would go from 2 to 5 to 32 in the logarithmic example which (once I add the conventions) gives us:

So, when you are given a logarithm to solve, just remember how to convert it to an exponential equation. Here are some practice problems, the answers are at the bottom.

## Properties of Logarithms[edit]

The following properities derive from the definition of logarithm.

If with , then for every real y,a,c it is:

1)

2)

3)

4)

There is also the "change of base rule":

for any

**Proof**

Let us take the log to base d of both sides of the equation :

.

Next, notice that the left side of this equation is the same as that in property number 1 above. Let us apply this property:

Isolating c on the left side gives

Finally, since

This rule allows you to evaluate logs to a base other than e or 10 on a calculator. For example,

- Solve these logarithms

- Evaluate with a calculator

- Find the
*y*value of these logarithms

## Answers[edit]

- Solve these logarithms
- 4
- 3
- 3

- Evaluate with a calculator
- 1.29248

- Find the
*y*value of these logarithms- 27
- 625
- 6561

Logarithms are the reverse of exponential functions, just as division is the reverse of multiplication, for example:

and

Or, in a more general form, if , then . Also, if , then , so if the two equations are graphed each is the reflection of the other over the line . (in both equations, *a* is considered to be the *base*).

Because of this, and .

Common bases used are bases of 10 which is a called a *common logarithm* or *e* which is called a *natural logarithm" (*e*~=2.71828182846).*

Common logs are written either as or simply as .

Natural logs are written either as or simply as (the ln stands for natural logarithm).

Logarithms are commonly abbreviated as logs.

## Properties of Logarithms[edit]

Proof:

and

and

and replace b and c (as above)

## Change of Base Formula[edit]

where *a* is any positive number, distinct from 1. Generally, *a* is either 10 (for common logs) or *e* (for natural logs).

Proof:

Put both sides to

Replace from first line

## Swap of Base and Exponent Formula[edit]

where a or c must not be equal to 1.

Proof:

by the change of base formula above.

Note that . Then

can be rewritten as

or by the exponential rule as

using the inverse rule noted above, this is equal to

and by the change of base formula