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Chapter 4: Problem 60

Find the domain of each logarithmic function. $$f(x)=\ln (x-7)^{2}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\ln (x-7)^{2}\) are all real numbers except \(x = 7\).

Step by step solution

01

Identify the argument of the logarithm

The argument of the logarithm in the function \(f(x)=\ln (x-7)^{2}\) is \((x-7)^2\). This is the value we need to be greater than zero since logarithms are undefined for arguments less than or equal to zero.
02

Find the domain of the argument

We must find the values of \(x\) such that \((x-7)^2>0\). However, any square of a real number is always non-negative (i.e., is either positive or zero). Therefore, the terms \((x-7)^2\) yields values greater than or equal to zero.
03

Exclude zero from the domain

Because the argument of a logarithm cannot be equal to zero, we must exclude the value \(x=7\) from the domain since that causes \((x-7)^2\) to equal zero. Hence, \(x\) can be any real number other than 7.

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Most popular questions from this chapter

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \ln x+\ln 7 $$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{3} x $$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{2}(x+2) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log N^{-6} $$
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