Introduction

Welcome to Topic 2.11! Today you'll learn what logarithmic functions are, why they behave the way they do, and how to work with their graphs and transformations.

Logarithmic Functions

A logarithmic function is defined as the inverse of an exponential function. 

For a given base $b>0,b\ne 1$, the equation $y={\log }_{b}x$ is equivalent to $b^y=x$. 

Using this, we can derive some of our basic logarithmic identities.

Additionally, because exponential functions have the domain $\left(-\infty ,\infty \right)$, and range $\left(0,\infty \right)$, and the logarithmic function is the inverse of the exponential function $\therefore$

the domain of ${\log }_{b}x$ is $\left(0,\infty \right)$ and the range is $\left(-\infty ,\infty \right)$

If $b>1$ , $\underset{x\to 0^{+}}{\lim }{\log }_{b}x=-\infty$, and when $0<b<1$, $\underset{x\to 0^{+}}{\lim }{\log }_{b}x=\infty$. 

Monotonicity and Concavity

Logarithmic functions are monotonic, meaning they are always increasing or always decreasing. Due to this, they also don’t have any extrema, with the exception of when its domain is restricted, in which case, the extrema of the function is its endpoints.

When $b>1$, the logarithmic function is always increasing on its domain, and when $0<b<1$, the logarithmic function is always decreasing on its domain.

Moreover, logarithms have the same concavity and no inflection points. If $b>1$, the graph is always concave down on its domain; if $0<b<1$, the graph is always concave up on its domain.

This means that when $b>1$, the logarithmic function is increasing at a decreasing rate, and when $0<b<1$, the logarithmic function is decreasing at an increasing rate.

Additive Transformations

Adding a constant to $x$ inside the logarithmic function shifts the graph horizontally, but does not change the shape. Similarly, if you add a constant to the output, it will simply shift the parent log function up or down.

Additive property of logs

A characteristic of logarithms is that equal increments in the output ($y$) correspond to constant multiplicative changes in the input ($x$). For instance, with $y={\log }_{2}x$, $x$ must double in order for $y$ to increase by one.

In general, if the inputs to a function are proportional over equal length output values, then the function must be logarithmic. Equivalently, if the outputs to a function are proportional over equal length input values, it would be exponential. Vertically or horizontally shifting the function does not modify this property.

Remember how to find how a function is exponential through a table? The outputs over equal length input intervals are proportional, right? Well, considering how exponentials are inverses of logarithms, you can actually do the opposite to figure out whether a function is logarithmic from a table! Functions that are logarithmic will have proportional inputs over constant interval outputs. 

Vertical and End Behavior

A logarithmic function $y={\log }_b\left(x+h\right)$ has a vertical asymptote at $x=-h$. Remember, if $b>1$ , $\underset{x\to h^{+}}{\lim }{\log }_{b}x=-\infty$, and when $0<b<1$, $\underset{x\to h^{+}}{\lim }{\log }_{b}x=\infty$.

Additionally, the behaviors are reversed when considering $x$ as it grows infinitely large. if $b>1$ , $\underset{x\to {\infty }^{+}}{\lim }{\log }_{b}x=\infty$, and when $0<b<1$, $\underset{x\to {\infty }^{+}}{\lim }{\log }_{b}x=-\infty$. Because of this, there are no horizontal asymptotes.

We see this occurring in exponential functions as well, except with horizontal asymptotes. In logarithms of ${\log }_b(x+h)$, there is a vertical asymptote at $x=-h$. For exponential functions of $b^x-h$, there is a horizontal asymptote at $y=-h$. This inverse relationship is important to know, as it is a big part of how exponential and logarithmic functions act along with many other relationships.

Practice Questions