Which of the following is equivalent to (log_(5)(m))/(log_(15)(m)) ? Choose 1 answer: (A) (1)/(3) (B) 3 (c) log(3) (D) log_(5)(15)

Question

Which of the following is equivalent to  (log_(5)(m))/(log_(15)(m)) ? Choose 1 answer: (A)  (1)/(3) (B) 3 (c)  log(3) (D)  log_(5)(15)

Bytelearn · Accepted Answer

Recognize relationship between logarithms: Recognize the relationship between the two logarithms. 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. Change of Base Formula: log_b(a) = log_c(a) / log_c(b)Apply change of base formula: Apply the change of base formula to the given expression. Using the change of base formula, we can express the given ratio as: (log_(5)(m))/(log_(15)(m)) = log(m) / log(5) / (log(m) / log(15))Simplify expression by dividing logarithms: Simplify the expression by dividing the logarithms. We can simplify the expression by dividing the numerators and denominators separately: (log(m) / log(5)) / (log(m) / log(15)) = (log(m) * log(15)) / (log(m) * log(5))Cancel out common terms: Cancel out the common terms. Since log(m) appears in both the numerator and the denominator, we can cancel it out: (log(m) * log(15)) / (log(m) * log(5)) = log(15) / log(5)Recognize remaining expression as constant: Recognize that the remaining expression is a constant. The expression 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. We know that 15 = 3 * 5, so we can use the product property of logarithms to expand log(15): log(15) = log(3 * 5) = log(3) + log(5) Now, we can substitute this into our expression: log(15) / log(5) = (log(3) + log(5)) / log(5)Simplify expression further: Simplify the expression further. We can separate the terms in the numerator: (log(3) + log(5)) / log(5) = log(3) / log(5) + log(5) / log(5) The second term, log(5) / log(5), simplifies to 1: log(3) / log(5) + 1Recognize log(3) / log(5) as constant: Recognize that log(3) / log(5) is a constant. 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.

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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Which of the following is equivalent to log⁡5(m)log⁡15(m) \frac{\log _{5}(m)}{\log _{15}(m)} log15​(m)log5​(m)​ ?\newlineChoose 111 answer:\newline(A) 13 \frac{1}{3} 31​\newline(B) 333\newline(C) log⁡(3) \log (3) log(3)\newline(D) log⁡5(15) \log _{5}(15) log5​(15)

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) $\log _{5}(15)$