# 4.4 Graphs of logarithmic functions

 Page 1 / 8
In this section, you will:
• Identify the domain of a logarithmic function.
• Graph logarithmic functions.

In Graphs of Exponential Functions , we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect .

To illustrate, suppose we invest $\text{\hspace{0.17em}}\text{}2500\text{\hspace{0.17em}}$ in an account that offers an annual interest rate of $\text{\hspace{0.17em}}5%,$ compounded continuously. We already know that the balance in our account for any year $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ can be found with the equation $\text{\hspace{0.17em}}A=2500{e}^{0.05t}.$

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? [link] shows this point on the logarithmic graph.

In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

## Finding the domain of a logarithmic function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as $\text{\hspace{0.17em}}y={b}^{x}\text{\hspace{0.17em}}$ for any real number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and constant $\text{\hspace{0.17em}}b>0,$ $b\ne 1,$ where

• The domain of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$
• The range of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(0,\infty \right).$

In the last section we learned that the logarithmic function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is the inverse of the exponential function $\text{\hspace{0.17em}}y={b}^{x}.\text{\hspace{0.17em}}$ So, as inverse functions:

• The domain of $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is the range of $\text{\hspace{0.17em}}y={b}^{x}:\text{\hspace{0.17em}}$ $\left(0,\infty \right).$
• The range of $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is the domain of $\text{\hspace{0.17em}}y={b}^{x}:\text{\hspace{0.17em}}$ $\left(-\infty ,\infty \right).$

Transformations of the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range of $\text{\hspace{0.17em}}y={b}^{x}.\text{\hspace{0.17em}}$ Similarly, applying transformations to the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ can change the domain . When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers . That is, the argument of the logarithmic function must be greater than zero.

For example, consider $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{4}\left(2x-3\right).\text{\hspace{0.17em}}$ This function is defined for any values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that the argument, in this case $\text{\hspace{0.17em}}2x-3,$ is greater than zero. To find the domain, we set up an inequality and solve for $\text{\hspace{0.17em}}x:$

In interval notation, the domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{4}\left(2x-3\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(1.5,\infty \right).$

the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard