Which of the following is equivalent to log(a)*log_(a)(5) ? Choose 1 answer: (A) log(a) (B) log(5) (c) log(5a) (D) log_(a)(5a)

Question

Which of the following is equivalent to  log(a)*log_(a)(5) ? Choose 1 answer: (A)  log(a) (B)  log(5) (c)  log(5a) (D)  log_(a)(5a)

Bytelearn · Accepted Answer

Understanding the expression: Understand the expression log(a)*log_(a)(5). The expression consists of two logarithms being multiplied together: log(a), which is the common logarithm of a, and log_(a)(5), which is the logarithm base a of 5.Recognizing applicable logarithm properties: Recognize the properties of logarithms that might apply. The multiplication of two logarithms does not directly correspond to any of the basic logarithm properties (product, quotient, or power rules). However, we can interpret log_(a)(5) as the power to which we must raise a to get 5.Applying the definition of logarithm base a of 5: Apply the definition of the logarithm base a of 5. By definition, log_(a)(5) = x means that a^x = 5. Therefore, log_(a)(5) is the exponent x.Substituting the definition back into the original expression: Substitute the definition back into the original expression. Since log_(a)(5) is the exponent x that makes a^x = 5, we can rewrite the expression as log(a) * x.Recognizing the expression cannot be further simplified: Recognize that the expression log(a) * x does not simplify further using common logarithm properties. The expression log(a) * x is already in its simplest form, given that x is the exponent that satisfies a^x = 5. There is no property of logarithms that allows us to combine a logarithm and an exponent in this way.Matching the expression to the given options: Match the expression to the given options. None of the options (A) log(a), (B) log(5), (C) log(5a), or (D) log_(a)(5a) are equivalent to the expression log(a) * x. Therefore, none of the options are equivalent to the original expression log(a)*log_(a)(5).

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

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