Logarithmic Functions

Once you are familiar with
logarithms and
exponential functions, you can look at logarithmic functions. Logarithms
are basically another way of writing
exponents and logarithmic functions are inverses of exponential functions.

An important definition that we should keep in mind is the definition of logs, because
it will help tremendously when dealing with logarithmic functions and various types
of log problems.

To reiterate, a logarithm is another way of writing an exponent. This definition
works in both directions (converting from exponential form to logarithmic and back).
The domain for logarithmic functions will be all positive real numbers for x
and the range will be all real numbers for y.

When we write log(x) without a base, it is understood that the base is 10.
On a calculutor, there are two types of logs, a base ten log and the natural log
ln(x). The natural log is of base e, which we found is a unique exponential
function. Many problems will deal with e and we will have to use the natural
log ln to evaluate and graph the function.

Graphing the exponential function and natural log function, we can see that they
are inverses of each other.

Let’s graph the function f(x) = log(x+2) of base 4.

We can use the definition of logs to rewrite this in exponential form. We can see
that the base is 4, the exponent is y, and the log will set to be
(x+2).

We can then plug in values for y and yield our x values. Though we usually plug
in x values to find our y values, it is much easier in this form to plug in various
values for y.

Notice that now we have a vertical asymptote at x = -2, and a point at (-1,0).
This is exactly the opposite of exponential functions, which have horizontal asymptotes
and a point at (0,1). All logarithmic functions will have a vertical asymptote and
pass through the point at distance 1 from the vertical asymptote in the direction
it opens up. This point will always be the x intercept. To find it, we can set y
equal to 0 and solve for x.

Let’s look at the graph of f(x) = ln(5-x).

Notice that our domain is x < 5. If our input is greater than 5, our output
will not be defined. The x intercept is given as

In the first step, we can use the definition of logs to rewrite the equation and solve
for x. Our x intercept is then (4,0).

Inverse Properties of Logarithms

Inverse Property I

This means that whenever the base of the log matches the base of the inside log,
the log will equal the exponent of the inside base. This is only if the bases match.

Inverse Property II

This means whenever the base raised to a log has the same base, then it simplifies
to whatever is inside the log. This is confirming that logs are another way of writing
exponents, just like subtracting is another way of writing addition and division
is another way of writing multiplication. Again, this property only works if the
the base b is the same as the base b of the log.

Let’s evaluate some logarithmic equations and expressions to practice our knowledge
of properties of logs.

Solve for x

Write as one logarithm

Solve for x