# Algebra II : Natural Log

## Example Questions

### Example Question #1 : Natural Log

Solve for .

Explanation:

The first thing we notice about this problem is that  is an exponent. This should be an immediate reminder: use logs!

The question is, which base should we choose for the log? We should use the natural log (log base e) because the right-hand side of the equation already has e as a base of an exponent. As you will see, things cancel out more nicely this way.

Take the natural log of both sides:

Rewrite the right-hand side of the equation using the product rule for logs:

Now rewrite the whole equation after bringing down those exponents.

is the same thing as , which equals 1.

Now we just divide by on both sides to isolate .

### Example Question #2 : Natural Log

Rewrite as a single logarithmic expression:

Explanation:

Using the properties of logarithms

and ,

we simplify as follows:

### Example Question #3 : Natural Log

Which of the following expressions is equal to the expression  ?

None of the other responses is correct.

Explanation:

By the reverse-FOIL method, we factor the polynomial as follows:

Therefore, we can use the property

as follows:

### Example Question #1 : Logarithms

Solve . Round to the nearest thousandth.

Explanation:

The original equation is:

Subtract  from both sides:

Divde both sides by :

Take the natural logarithm of both sides:

Divde both sides by  and use a calculator to get:

### Example Question #5 : Natural Log

What are the domain and the range of the function ?

Domain = all positive numbers

Range = all positive numbers

Domain = all positive real numbers

Range = all real numbers

Domain = all positive numbers

Range = all non-negative numbers

Domain = all real numbers

Range = all real numbers

Domain = all non-negative numbers

Range = all positive numbers

Domain = all positive real numbers

Range = all real numbers

Explanation:

Remember that  is still a logarithm of a positive number, .

It's not possible to raise  to ANY power and obtain a negative number. Because even , for example, is just , which is a ratio of two positive numbers, and therefore positive.

More than that, it's also not possible to obtain 0 by raising  to any power. Think: "To what power can I exponentiate e and obtain 0?"

So the domain is strictly positive. It excludes negative numbers and 0.

What about the range? To what possible values are we allowed to exponentiate ?

Well, we just saw that  has a definition for negative numbers.  (this fact is true for ALL numbers, not just ).

And we can obviously raise it to positive powers. So the range is all real numbers. It includes negative numbers, 0, and positive numbers.

### Example Question #2 : Logarithms

Solve for :

.

If necessary, round to the nearest tenth.

No solution

Explanation:

Give both sides the same base, using e:

.

Because e and ln cancel each other out, .

Solve for x and round to the nearest tenth:

### Example Question #7 : Natural Log

Solve for x:

Explanation:

To solve for x, keep in mind that the natural logarithm and the exponential cancel each other out (property of any logarithm with a base that is being taken of that same base with an exponent attached). When they cancel, we are just left with the exponents:

### Example Question #1 : Natural Log

Determine the value of:

Explanation:

The natural log has a base of .  This means that the term will simplify to whatever is the power of .  Some examples are:

This means that

Multiply this quantity with three.

### Example Question #9 : Natural Log

Determine the value of:

Explanation:

In order to simplify this expression, use the following natural log rule.

The natural log has a default base of . This means that:

Simplify: