Which of the following is equivalent to $\frac{\log _{5}(m)}{\log _{15}(m)}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{3}$ $\newline$ (B) $3$ $\newline$ (C) $\log (3)$ $\newline$ (D) $\log _{5}(15)$

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Algebra 2

Product property of logarithms

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Q. Which of the following is equivalent to

\frac{\log _{5}(m)}{\log _{15}(m)}

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{3}

\newline

(B)

3

\newline

(C)

\log (3)

\newline

(D)

\log _{5}(15)

Recognize relationship between logarithms: Recognize the relationship between the two logarithms. $\newline$ The given expression is a ratio of two logarithms with different bases but the same argument $m$ . We can use the change of base formula to rewrite the expression in terms of logarithms with a common base. $\newline$ Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$
Apply change of base formula: Apply the change of base formula to the given expression. $\newline$ Using the change of base formula, we can express the given ratio as: $\newline$ $\frac{\log_{5}(m)}{\log_{15}(m)} = \frac{\log(m)}{\log(5)} / \frac{\log(m)}{\log(15)}$
Simplify expression by dividing logarithms: Simplify the expression by dividing the logarithms. $\newline$ We can simplify the expression by dividing the numerators and denominators separately: $\newline$ $(\frac{\log(m)}{\log(5)}) / (\frac{\log(m)}{\log(15)}) = \frac{\log(m) \cdot \log(15)}{\log(m) \cdot \log(5)}$
Cancel out common terms: Cancel out the common terms. $\newline$ Since $\log(m)$ appears in both the numerator and the denominator, we can cancel it out: $\newline$ $\frac{\log(m) \cdot \log(15)}{\log(m) \cdot \log(5)} = \frac{\log(15)}{\log(5)}$
Recognize remaining expression as constant: Recognize that the remaining expression is a constant. $\newline$ The expression $\frac{\log(15)}{\log(5)}$ is a constant because it does not depend on the variable $m$ . We can calculate this constant using the properties of logarithms.
Calculate value of constant: Calculate the value of the constant. $\newline$ We know that $15 = 3 \times 5$ , so we can use the product property of logarithms to expand $\log(15)$ : $\newline$ $\log(15) = \log(3 \times 5) = \log(3) + \log(5)$ $\newline$ Now, we can substitute this into our expression: $\newline$ $\frac{\log(15)}{\log(5)} = \frac{(\log(3) + \log(5))}{\log(5)}$
Simplify expression further: Simplify the expression further. $\newline$ We can separate the terms in the numerator: $\newline$ $(\log(3) + \log(5)) / \log(5) = \log(3) / \log(5) + \log(5) / \log(5)$ $\newline$ The second term, $\log(5) / \log(5)$ , simplifies to $1$ : $\newline$ $\log(3) / \log(5) + 1$
Recognize $\log(3) / \log(5)$ as constant: Recognize that $\log(3) / \log(5)$ is a constant. $\newline$ The term $\log(3) / \log(5)$ is a constant and cannot be simplified further without a calculator. However, we can see that none of the answer choices match this form. We need to re-evaluate our steps to see if we made a mistake.