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

what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
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Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
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Abhi
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salma
Commplementary angles
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Nharnhar
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a perfect square v²+2v+_
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