Logarithmic Functions

Definition:

Let a > 0, a ≠ 1. Then  log_ax 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 log₁₀16 = 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 log₁₀16 = 2 .

Example: Show that log₁₀10,000 = 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 log₁₀10,000 = 4.

Example: log₁₀.001 = ?

Here the base is 10, and x = .001 . Write .001 as a power of 10. Since
.001 = 10^-3 ,log₁₀.001 = -3

The function log₁₀ 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) = ax	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