4.4 Graphs of logarithmic functions

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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 $\text{\hspace{0.17em}}\left(1.5,\infty \right).$

give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena