Algebra/Logarithms

From Wikibooks, open books for an open world
Jump to: navigation, search

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.

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