4.3 Graphs of logarithmic functions  (Page 2/8)

Graphing logarithmic functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function $y={\log }_b\left(x\right)$ along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function $y={\log }_b\left(x\right).$ Because every logarithmic function of this form is the inverse of an exponential function with the form $y=b^x,$ their graphs will be reflections of each other across the line $y=x.$ To illustrate this, we can observe the relationship between the input and output values of $y=2^x$ and its equivalent $x={\log }_2(y)$ in [link] .

Using the inputs and outputs from [link] , we can build another table to observe the relationship between points on the graphs of the inverse functions $f(x)=2^x$ and $g(x)={\log }_2(x).$ See [link] .

As we’d expect, the x - and y -coordinates are reversed for the inverse functions. [link] shows the graph of $f$ and $g.$

Observe the following from the graph:

• $f(x)=2^x$ has a y -intercept at $(0,1)$ and $g(x)={\log }_2\left(x\right)$ has an x - intercept at $(1,0).$
• The domain of $f(x)=2^x,$ $\left(-\infty ,\infty \right),$ is the same as the range of $g(x)={\log }_2\left(x\right).$
• The range of $f(x)=2^x,$ $\left(0,\infty \right),$ is the same as the domain of $g(x)={\log }_2\left(x\right).$

A General Note

Characteristics of the graph of the parent function, f ( x ) = log _b ( x )

For any real number $x$ and constant $b>0,$ $b\ne 1,$ we can see the following characteristics in the graph of $f(x)={\log }_b\left(x\right):$

• one-to-one function
• vertical asymptote: $x=0$
• domain: $(0,\infty )$
• range: $\left(-\infty ,\infty \right)$
• x- intercept: $(1,0)$ and key point $(b,1)$
• y -intercept: none
• increasing if $b>1$
• decreasing if $0<b<1$

See [link] .

[link] shows how changing the base $b$ in $f(x)={\log }_b\left(x\right)$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. ( Note: recall that the function $\ln \left(x\right)$ has base $e\approx \text{2}.\text{718.)}$