Introductory Calculus: Exponentials and Logarithms

By M Ransom

Properties of Exponents: To review some of the fundamental properties of exponents, we will look at several examples.

If a > 0, b > 0, and x and y are any real numbers, then:

a^x ● a^y = a^x+y

Properties of Logarithms: To review some of the fundamental properties of logarithms, we will look at several examples.

If x and y are both positive numbers, then:

Meaning of Log: If .

If

Notice: the log is the exponent.

log(x) + log(x + 2) = log(x² + 2x)

Notice: if the base b is not shown, the base is assumed to be 10.

ln(x) means log_e(x) where .

Putting It All Together:

Solve for x if .

By direction inspection we can see that x must equal 4.

Solve for x if .

In this case, direct inspection fails us since 4 is too big - giving us 16 - and 3 is too small - only giving us 8.

To get an exact answer, we would use logarithms:

Step 1:
Step 2:
Step 3:

Let's Practice. In the following five equations, solve for x (accurate to 3 decimal places).

Graphs: A graph of both

is shown below right. Notice that in order to graph

, it is necessary to rewrite as

since there is no log base 2 button on the calculator. Also, notice that the point (1, 2) is on

. Since

is its inverse, the point (2, 1) will be on that graph.

Derivatives: Given below are ten examples of taking derivatives of exponential functions and logarithms.

Show that has a relative minimum at .

We know from example #3 above that .

If we set this equal to 0, we get .

We note that for , this derivative is negative, and for , the derivative is positive.

This change in sign from negative to positive is the proof that we need to know that has a minimum value when .