# The Natural Logarithm

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.

1. . (This makes sense, since .)
2. .
3. .
4. .

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 .

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

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