Graphs of logarithmic equations of the form \\ y=a\log_{B}(x-h) + k (where a,\,B, and h and k are any number and B>0) are called logarithmic graphs.

This is the logarithmic graph of y=\log_{2}x.

Like lines, logarithmic graphs will always have an x-intercept. This is the point on the graph which touches the x-axis. We can find this by setting y=0 and finding the value of x. For example, the x-intercept of y=\log_{2}x is (1,0).

Similarly, we can look for y-intercepts by setting x=0 and then solving for y. Because this is a logarithmic equation, there could be 0 or 1 solutions, and there will be the same number of y-intercepts. For example, the graph of y=\log_{2}x has no y-intercept.

Logarithmic graphs have a vertical asymptote which is the vertical line which the graph approaches but does not touch. For example, the vertical asymptote of y=\log_{2}x is x=0.

Examples

Example 1

Consider the function y = \log_{4} x.

Complete the table of values for y = \log_{4} x, rounding any necessary values to two decimal places.

x	0.3	1	2	3	4	5	10	20
y	-0.87			0.79			1.66

Worked Solution

Create a strategy

Use the change of base law: \log_{a} b = \dfrac{\log_{10} b}{\log_{10} a}

Apply the idea

Substitute each of the x-values in the table into the equation and evaluate.

For x=1:

\displaystyle y	\displaystyle =	\displaystyle \log_{4} x
	\displaystyle =	\displaystyle \log_{4} 1	Substitute x=1
	\displaystyle =	\displaystyle \dfrac{\log_{10} 1}{\log_{10} 4}	Use the change of base law
	\displaystyle =	\displaystyle 0	Evaluate

Similarly, by substituting the remaining x-values into \log_{4} x, we get:

x	0.3	1	2	3	4	5	10	20
y	-0.87	0	0.5	0.79	1	1.16	1.66	2.16

Which of the following is the graph of y = \log_{4} x?

-4

-3