Let a be a positive number, , and let . The * logarithm of x to the base a* is the number such that . That is,

* Example.* What exponential equation is
equivalent to ?

* Example.* What logarithmic equation is
equivalent to ?

* Example.*

Remember that e is the number whose decimal value is . According to the definition above,

Logs to the base e are so important that they have a special name;
they are called * natural logarithms*. They also
have a special symbol: Write in place of .

On most calculators, and are on the same button. For example, you can use your calculator to verify that

The graph of the natural logarithm is shown below:

and are * inverse functions* --- if
you do one, then the other, you get what you started with. In
symbols,

These relationships are often useful for solving equations involving or .

In addition, satisfies the usual properties of logarithms.

* Properties of the natural logarithm.*

- . (This makes sense, since .)
- .
- .
- .

* Example.* Solve .

To get something out of an exponent, take logs:

* Example.* Solve .

To get something out of a log, use :

Check the possible solutions by plugging back in:

Therefore, the only solution is .

* Example.* $1000 is to be invested at 4%
annual interest, compounded quarterly. For how many years must the
investment be held to accrue to at least $10000?

Let n be the number of years required. Then

* Example.* Write in terms of , , and .

* Example.* Write as a
single log.

* Example.* Solve .

To get something out of an exponent (the ), take logs:

Then

The solution is .

* Example.* Solve .

Take logs on both sides:

Then

And so

The solution is .

* Example.* How many years will it take $5000
invested at annual interest, compounded monthly, to accrue to
$10000?

Plug , , , in the compound interest formula

I get

Then

Take logs on both sides:

So

Thus, years.

* Example.* Solve .

means that . Therefore,

The base of a logarithm must be positive, so .

* Example.* Compute on your
calculator.

Your calculator can compute logs to the base 10 and natural logs, which are logs to the base e. To compute logs to other bases, use the conversion formula

You t ake c to be a base available on your calculator. For example, using natural logs,

So in this case,

* Example.* Solve .

Then

Factor and solve:

Check: If ,

But if , is undefined.

Hence, the only solution is .

* Example.* Solve for x: .

Then

Factor and solve:

Check: and can't be substituted into the original equation, because you can't take the log of 0 or a negative number.

If ,

The only solution is .

* Example.* Solve for x: .

Since , the equation is

Factor and solve:

is impossible, because for all x.

To solve , take logs of both sides:

The only solution is .

* Example.* How much must be invested at 3.6%
annual interest, compounded monthly, to accrue to $3000 after 2
years?

Using the compound interest formula,

Therefore,

* Example.* If and , compute .

* Example.* Find the domain of .

You can't take the log of 0 or a negative number, so f is only defined if .

is a parabola opening upward with roots at and .

The domain is or .

* Example.* Solve for x: .

Factor and solve:

Solve the two equations by exponentiating:

The solutions are and .

* Example.* Solve for x: .

Then

Then

Copyright 2008 by Bruce Ikenaga