Try our Free Online Math Solver!

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
Prev  Next 