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Chapter 3: Problem 99

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right)$$

Short Answer

Expert verified
The given equation \( \log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right) \) is True.

Step by step solution

01

Identify and Apply the Logarithm Rule

By the quotient rule of logarithms, we know that \( \log_b(M/N) = \log_b(M) - \log_b(N) \). In the given equation, the left-hand side is in the format of the quotient rule. Hence, it can be rewritten as:\[ \log _{6}(x-1)-\log _{6}(x^{2}+4)\]
02

Compare Both Sides of Given Equation

Now, compare this result with the right-hand side of the given equation. Upon comparison, we can see these expressions are equivalent.
03

Final Conclusion

Given that both sides are equivalent, we can safely conclude that the given statement is true as it correctly applies the quotient rule of logarithms.

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