The Logarithm Function
You should be familiar with exponential functions.
You should also be familiar with inverse functions
and how the graph of the inverse function
relates to the graph of the original function.
In this lesson, we will define the inverse
to the exponential function, called the logarithm
function. We will then rewrite exponential
equations using the logarithmic notation and
look at properties of the graph of the logarithm
function.
Let’s begin with the exponential function
with base 10, that is, the function y = 10^x.
Anything to the zero power is 1, so when
the exponent is 0, the y-value is 1. When
the exponent is 1, the y-value is 10. Exponential
growth is rapid, when the exponent is 12,
the y-value is one trillion. Negative exponents
are reciprocals, so when the exponent is negative,
the y-value is a fraction less than 1, but
always positive. The graph of this exponential
function increases from left to right, passes
through the point (0, 1), is asymptotic to
the x-axis on the negative side (the left),
bends upward, has all real numbers for its
domain and all positive numbers for its range.
We now wish to find an inverse to the exponential
function, that is, we wish to find a function
which has as it’s input various powers of
10, and has as its output, the exponent. This
inverse to the exponential function is called
the logarithm function. In this case, since
the base of the exponential function was 10,
we call this the base 10 logarithm.
Recall that an inverse function reverses the
roles of x and y, thus the inverse of the
exponential function has y as the exponent.
The notation for logarithms looks like this.
We will often rewrite logarithmic equations
as exponential equations. Remember that y,
the log, is the exponent.
What is the base 10 log of 1000? Let’s write
this in exponential form. The log is the exponent,
so y is the exponent. The base is 10. That
leaves only one place to put the number 1000
in the exponential equation. y is the exponent
you put on 10 to get 1000. That exponent is
3.
Here again is the graph of the exponential
function y = 10^x . We get the graph of the
logarithmic function by reflecting this across
the line y = x. Each of the properties we
had for the exponential function now reverses
the roles of x and y. The exponential function
went through the point (0, 1), so the log
function goes through the point (1, 0). The
exponential function approached the negative
half of the x-axis from above, that is, from
the positive side. The log graph approached
the negative half of the y-axis from the positive
side, that is, from the right. The domain
of the exponential function is the range of
the log function, and vice versa. The exponential
graph bends sharply upward (the positive y-direction),
so the log graph bends sharply in the positive
x-direction, that is, to the right.
Let’s practice some logarithms with other
bases, rewriting them in exponential form.
Recall that the log is the exponent, so in
the first equation, 3 is the exponent. The
base is 2, so we write 2^3 = 8. You may wish
to pause the video to rewrite the second equation
in exponential form.
