Exponents and Logarithms

Study Exercises

Complete odd-numbered problems 1–23 in the Written Exercises section on pages 189–190 of your
text. Then check your answers in the back of the text.

Section 5-5: Logarithmic Functions pp. 191–197

If I needed to solve the problem 18 = 2^x, I would need some way to solve for x. Normally I would
use the inverse function. For example, let's examine how we solve 2 x + 3 = 13.

The inverse of addition is subtraction :

The inverse of multiplication is division:

For the problem 18 = 2^x, however, the four basic operations will not get the x alone. I need an
inverse function for raising to a power. That inverse function is the logarithmic function. A
logarithmic function is the inverse of an exponential function . By definition, x = b^a is

Notice the exponent, a, is alone in the log function. In log_b x = a, the b is called the base. Even
x = e^a can be written as log_ex = a. Not surprisingly, the log of base e has been given a special name,
the natural logarithm function. And it has a special look, log_ex = a -> ln x = a. The log base e is
replaced with natural log, ln. Let's do a little work with converting from log form to exponential
form.

Example 1: Write log₃81 = 4 in exponential form:

Example 2: Write in exponential form.

We are missing the value of the log. I will set equal to x:

Now I will take it a step further and de termine the value of x:

So the value of the log, −3 , is the exponent of 5 (the base). That gives you 1/125

Example 3: Write in exponential form and find its value.

Write in exponential form:

The cube root is the power of 1/3:

So the value of the log, −1/3, is the exponent of 6 (the base). That gives you

Example 4: Let's try problem 18 on page 195 of your text.

(a)
When a base is not given, the is base 10.

It looks like when the base is raised to a power, the power is the value of the log:

So this would have a value of 8.

Remember ln is log_e

Example 5: Find

Start with log₄16. That has a value of 2. So now find 3² = 9.

Example 6: Find

Again, Notice the 64 in the problem. So b log_bx = x.

Example 7: If

Swap the x and y:

Solve for y using log function; write in log form:

Example 8: Let's try problem 34 on page 195 of your text.

The domain involves values placed in for x. You can never take the log of a negative number
or zero, because raising a number to a power will never make the number zero or negative . So
x − 2 > 0. The domain is {x| x > 2}. The range involves the values of y. Keeping in mind the
value of a log is the power the base is raised to, y can be any number. You can raise numbers
to negative powers, fractional powers , positive powers, and zero. The range is all real
numbers. The zeros would be found like this:

Rewrite in exponential form:

The zero is 3.

Here you are taking the log of x only. The domain is {x| x > 0}. The range is all real numbers .
Zeros are found in this way:

Rewrite in exponential form:

The zero is 4.

Example 9: Solve for x:

Write in exponential form; this log has a base of 10:

Example 10: Solve for x:

Let's replace log₂ xwith y:

Write in exponential form:

So now we know log₂ x = 6.
Rewrite log₂ x = 6 in exponential form:

Example 11: Solve ln (x − 2) = 1.

Write in exponential form:

Example 12: Solve ln x = − 0.52 using your calculator.
Written in exponential form:

Study Exercises

Complete odd-numbered problems 1–49 in the Written Exercises section on pages 194–197 of your
text. Then check your answers in the back of the text.

Section 5-6: Laws of Logarithms pp. 197–203

Let's experiment with log₂ 4 + log₂ 8 = log₂ 32. Is it a true statement?

So it would appear that to add logs of like bases, you multiply numbers you are taking logs of. This
is kind of the inverse of It is a multiplication problem, and you add exponents. So in
, would you divide because in you subtract exponents?

Yes, you divide the 4 and 8 to get an equivalent log for the difference.

What about That would be log₂ 64, which is 6. That is three times the value of log₂ 4, 2. So
could be written as 3 log₂ 4. These observations are written as laws on page 197 of your text.
These laws can be used to simplify log problems and make work easier.

Example 1: Let's try problem 4 on page 200 of your text:

Using law number 1:

Using law number 4:

or

Example 2: Rewrite in terms of log x:

Using law number 4:

or

Example 3: Let's try problem 10 on page 200 of your text:

Using law number 4:

Using law number 2:

Example 4: Let's try problem 16 on page 200 of your text:

Using law number 1:

Using law number 2:

Using law number 4:

Example 5: Simplify

Let's work with ln. That is, e to some power is . Now in our original problem, e is taken to
that power. So it now has a value of .

Example 6: Simplify

Let's simplify 1 + 2 log x as one log. The base is 10 so log 10 = 1.

This means 10 to some power is 10 x^2:

Now back to the original problem:

Now 10 is taken to the power mentioned above. So it has a value of 10 x^2.

Example 7: Express in terms of x, ln y = 2 ln x − ln 6:

Example 8: Let's try problem 34b on page 200 of your text: ln y = 3 − 0.5 x
This does not have a log on both sides. I will write it in exponential form:

Example 9: Let's try problem 38 on page 201 of your text.

(b)
Horizontally stretching the graph by a factor of 3 is equivalent to shifting down one unit. The
f (x), which is y on the graph, has been decreased by 1.

Example 10: If log₉ 5 = x and log₉ 4 = y, express in terms of x and y:

Example 11: Solve log₅ (x + 24) + log₅ x = 2.

You need a log on the right side with a base of 5. The log would need to be equivalent to a power of 2.

The value of x cannot be −25 because you can't take the log of a negative number or zero. This
leaves 1 as the solution. Let's check it:

Example 12: Let's try problem 50 on page 202 of your text:

Convert 3 to log₂ 8:

But the value of −6 and less will not work. The solution is x ≥ 3.

Study Exercises

Complete odd-numbered problems 1–53 in the Written Exercises on pages 200–203 of your text.
Then check your answers in the back of the text.

Section 5-7: Exponential Equations; Changing Bases pp. 203–207

Now we will use our logarithms to help us find the value of an exponent. In Section 5-2, we found
the values of an exponent by rewriting the problem with like bases:

But not all problems can be rewritten in the same base. In these cases the log function will be used in
the problem and our trusty calculator will find the value of the logs.

Example 1: Solve 25^x = 2.
Place the log function on both sides of the equation:

Use law number 4:

Divide to get x alone:

Your calculator has a log key, and it is set for base 10. Base 10 is also the base in this problem. Use
your calculator to find both log values:

To the nearest hundredth is .22.
Example 2: Solve

Example 3: Let's try problem 14 on page 205 of your text.

Use is the original amount, when it is tripled it will look like

Divide P₀ over and it's gone:

Use the natural log:

It will take 15.7 years or 15 years and 8.4 months to triple the investment.
As I mentioned earlier, your calculator is set up for base 10. But that won't stop you from being able
to solve problems in other bases. There is a change-of-base formula:

Example 4: Solve 5^x = 98.
Rewrite in log form:

Use the change-of-base formula:

Example 5: Let's try problem 24 on page 206 of your text:

If you have a graphing calculator and you would like to graph the change to , here is the
procedure to use. Since y and x are to be the same also, graph y = x on the same coordinate axis.
Keep zooming in until you can trace and find the point of intersection to the nearest hundredth. It
will have a value of 1.85.

Example 6: Let's try problem 30 on page 206 of your text:

Multiply both sides by e^x:

Replace e^x with y:

Use the quadratic formula :

Replace y with e^x:

Rewrite in log form:

Example 7: Let's try problem 36 on page 207 of your text.
Replacing log_ab and log_bc with the change-of-base formula:

Rewrite as a single log using change-of-base formula: log_ac.

Example 8: Let's try problem 40 on page 207 of your text.
Using the proven formula from problem 35:

Using law number 1:

Study Exercises
Complete odd-numbered problems 1–43 in the Written Exercises section on pages 205–207 of your
text. Then check your answers in the back of the text.