# Logarithmic Functions

**Definition:**

Let a > 0, a ≠ 1. Then log_{a}x is the number to which you raise a to get x.

Logarithms are in essence exponents. Their domains are powers of the base and

their ranges are the exponents needed to produce those particular powers.

Example: Demonstrate that
log_{10}16 = 2 .

Here the base is 4 and x = 16. To what number do you have to raise 4

in order to get 16? Answer: 2, so
log_{10}16 = 2 .

Example: Show that
log_{10}10,000 = 4 .

Here the base is 10, and x = 10,000. What number do you have to raise

10 to, in order to get 10,000 (4 zeros )? Answer: 4, so
log_{10}10,000 = 4.

Example:
log_{10}.001 = ?

Here the base is 10, and x = .001 . Write .001 as a power of 10. Since

.001 = 10^{-3} ,log_{10}.001 = -3

The function
log_{10} x is known as the common logarithm. Sometimes

you will see the expression log x .

**Theorem:**

Let a > 0, a ≠ 1 . Then
log_{a} x and a^{x} are inverse to each other.

Remark:

This theorem is too difficult to prove here. However, if you let f (x) = a^{x} and

you can show the following.

a)
What does this mean? It is a, raised to the

number to which you raise a to get x. So it equals x, that is

b)

So we haven’t completely proved this theorem, but you can see that
log_{a} x and

a^{x} undo each other.

Domain of f | f (x) = a^{x} |
Range of f |

Range of g | Domain of g |

** Graphs of Logarithmic Functions:
**Knowing that
log

_{a}xand a

^{x}are inverses allows you immediately to graph

log

_{a}x If you wish to graph the function you need only graph the

function g(x) = 2

^{x}, and flip it around the line y = x

** Solving Exponential Equations**

We are now able to solve exponential equations by “getting at” the

exponent of a term .

Example: Solve for x: 2^{x} = 3

Since one can’t write 3 as a power of 2, this rather simple problem can’t be

solved without logarithms. But now, knowing that
is the inverse of 2^{x}

one simply “ Logs both sides.”

You see that logarithms isolate the exponent .

Example: Solve of x:

First log both sides:

And now take the square root of both sides so that

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