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Chapter 4: Problem 3

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. $$ \log _{7}(7 x) $$

Short Answer

Expert verified
The expanded form of the given logarithmic expression is \(\log_{7}(x) + 1\).

Step by step solution

01

Apply Logarithmic Product Rule

Firstly, apply the logarithmic product rule which indicates that \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\). In this case our \(m\) is \(7\), and \(n\) is \(x\), hence the expression is equivalent to \(\log_{7}(7) + \log_{7}(x)\).
02

Simplify \(\log_{7}(7)\)

Simplify the first of the terms by using the property that \(\log_{b}(b) = 1\) for any positive \(b \neq 1\). So, \(\log_{7}(7)\) simplifies to \(1\). This leaves the result as \(1 + \log_{7}(x)\).
03

Rearrange the Terms

For a clean representation and more standard appearance, the terms can be rearranged as \(\log_{7}(x) + 1\). It's the final expanded form of the logarithmic expression given in the exercise.

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Most popular questions from this chapter

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

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. $$ \log _{5} \sqrt[3]{\frac{x^{2} y}{25}} $$

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. $$ \log _{9}(9 x) $$

Which one of the following is true? a. If \(\log (x+3)=2,\) then \(e^{2}=x+3\) b. If \(\log (7 x+3)-\log (2 x+5)=4,\) then in exponential form \(10^{4}=(7 x+3)-(2 x+5)\) c. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\) d. Examples of exponential equations include \(10^{x}=5.71, e^{x}=0.72,\) and \(x^{10}=5.71\)

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. $$ \log (10,000 x) $$
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