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Chapter 12: Problem 7

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{7}\left(\frac{7}{x}\right)$$

Short Answer

Expert verified
The simplified form of \( \log _{7}\left(\frac{7}{x}\right) \) is \( 1 - \log_7(x) \).

Step by step solution

01

Apply the Quotient Rule

Rewrite the given expression \( \log _{7}\left(\frac{7}{x}\right) \) using the quotient rule of logarithms which states that \( \log_b(M/N) = \log_b(M) - \log_b(N) \). Thus, this expression becomes \( \log_7(7) - \log_7(x) \).
02

Simplify Using the Property of Logarithms

Now, use the property \( \log_b(b) = 1 \), to simplify \( \log_7(7) \) to '1'. Hence, the expression further simplifies to \( 1 - \log_7(x) \).
03

Final Simplified Expression

After simplification, the expanded expression of \( \log _{7}\left(\frac{7}{x}\right) \) is \( 1 - \log_7(x) \).

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$

Solve: $$\frac{3}{x+1}-\frac{5}{x}=\frac{19}{x^{2}+x}$$

Describe the product rule for logarithms and give an example.

a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$
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