Exponential Functions

Exponential functions have the form    f(x) = b^x

where b is the base and x is the exponent (or power).

If b is greater than 1, the function continuously increases in value as x increases. A special property of exponential functions is that the slope of the function also continuously increases as x increases.

Other calculators have a button labeled x^y which is equivalent to the ^ symbol.It is common to write exponential functions using the carat (^), which means "raised to the power". Computer programing uses the ^ sign, as do some calculators.

Example of an Exponential Function

Consider the function f(x) = 2^x.

In this case, we have an exponential function with base 2. Some typical values for this function would be:

x	-2	-1	0	1	2	3
f(x)	1/4	1/2	1	2	4	8

Here is the graph of y = 2^x.

Notice:

• That as x increases, y also increases.
• That as x increases, the slope of the graph also increases.
• That the curve passes through (0, 1). All exponential curves of the form f(x) = b^xpass through (0, 1), if b > 0.
• The curve does not pass through the x-axis. It just gets closer and closer to the x-axis as we take smaller and smaller x-values.

Logarithmic Functions

A logarithm is simply an exponent that is written in a special way.

For example, we know that the following exponential equation is true:

3² = 9

In this case, the base is 3 and the exponent is 2. We can write this equation inlogarithm form (with identical meaning) as follows:

log₃9 = 2

We say this as "the logarithm of 9 to the base 3 is 2". What we have effectively done is to move the exponent down on to the main line. This was done historically to make multiplications and divisions easier, but logarithms are still very handy in mathematics.

The logarithmic function is defined as:

f(x) = log_bx

The base of the logarithm is b.

The 2 most common bases that we use are base 10 and base e, which we meet inLogs to base 10 and Natural Logs (base e) in later sections.

The logarithmic function has many real-life applications, in acoustics, electronics, earthquake analysis and population prediction.

Example 1:

Write in logarithm form: 8 = 2³

Answer

log₂8 = 3
This just follows from the definition of a logarithm.

Example 2:

Write in exponential form: log₁₀1000 = 3

Answer

1000 = 10³
Once again, this just follows from the definition of a logarithm.

Example 3:

Find b if

Answer

The first line follows from the logarithm definition. The second line uses negative exponents.
Then we find the 4th root of both sides.

Exercises

1. Evaluate y = 9^x if x = 0.5

Answer

y = 9^0.5
y = 3

This is just a fractional index.

2. Express 8² = 64 in logarithmic form.

Answer

log₈64 = 2

3. Express log₁₁121 = 2 in exponential form.

Answer

11² = 121

4. Determine the unknown: log₁₀ 0.01 = x

Answer

Writing this in exponential form, we have: 10^x = 0.01. Solving this gives:

x = -2

5. Determine the unknown: log_b (1/4) = -1/2

Answer

Using the log laws, we can write this as:

So, using the rule for negative exponents gives:

Squaring both sides gives: b = 16.
Checking our answer, we have:
And as a logarithm, this can be written as:

log₁₆(1/4) = −1/2