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Let a > 0, a ≠ 1. Then logax is the number to which you raise a to get x.
Example: Demonstrate that log1016 = 2 .
Example: Show that 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
log10 x is known as the common logarithm. Sometimes
you will see the expression log x .
Let a > 0, a ≠ 1 . Then loga x and ax are inverse to each other.
This theorem is too difficult to prove here. However, if you let f (x) = ax and
you can show the following.
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
So we haven’t completely proved this theorem, but you can see that
loga x and
ax undo each other.
|Domain of f||f (x) = ax||Range of f|
|Range of g||Domain of g|
Graphs of Logarithmic Functions:
Knowing that loga xand ax are inverses allows you immediately to graph
loga x If you wish to graph the function you need only graph the
function g(x) = 2x , and flip it around the line y = x
Solving Exponential Equations
Example: Solve for x: 2x = 3
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