Evaluate: log_(64)16 Answer:

Identify Relationship: Identify the relationship between the base of the logarithm and the number inside the logarithm. We have log base 64 of 16, which means we are looking for the power to which 64 must be raised to get 16. We know that 64 is 2 raised to the 6th power (2^6) and 16 is 2 raised to the 4th power (2^4).Express as Powers: Express both the base and the number inside the logarithm as powers of a common base. 64 = 2^6 and 16 = 2^4. So, log base 64 of 16 can be written as log base (2^6) of (2^4).Apply Change of Base: Apply the change of base formula for logarithms. The change of base formula states that log base (a) of (b) can be written as (log base (c) of (b)) / (log base (c) of (a)), where c is any positive number. Using 2 as the new base, we get (log base 2 of 16) / (log base 2 of 64).Evaluate Using Common Base: Evaluate the logarithms using the common base 2. log base 2 of 16 is 4 because 2^4 = 16. log base 2 of 64 is 6 because 2^6 = 64. So, we have (4) / (6).Simplify Fraction: Simplify the fraction. (4) / (6) simplifies to 2 / 3. Therefore, log base 64 of 16 is equal to 2 / 3.

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Evaluate: $\newline$ $\log _{64} 16$ $\newline$ Answer:

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Precalculus

Product property of logarithms

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Q. Evaluate:

\newline

\log _{64} 16

\newline

Answer:

Identify Relationship: Identify the relationship between the base of the logarithm and the number inside the logarithm. $\newline$ We have $\log_{64} 16$ , which means we are looking for the power to which $64$ must be raised to get $16$ . $\newline$ We know that $64$ is $2$ raised to the $6$ th power ( $2^6$ ) and $16$ is $2$ raised to the $4$ th power ( $64$ $0$ ).
Express as Powers: Express both the base and the number inside the logarithm as powers of a common base. $\newline$ $64 = 2^6$ and $16 = 2^4$ . $\newline$ So, $\log_{64} 16$ can be written as $\log_{(2^6)} (2^4)$ .
Apply Change of Base: Apply the change of base formula for logarithms. The change of base formula states that $\log_{a}(b)$ can be written as $\frac{\log_{c}(b)}{\log_{c}(a)}$ , where $c$ is any positive number. Using $2$ as the new base, we get $\frac{\log_{2}(16)}{\log_{2}(64)}$ .
Evaluate Using Common Base: Evaluate the logarithms using the common base $2$ . $\log_{2} 16 = 4$ because $2^{4} = 16$ . $\log_{2} 64 = 6$ because $2^{6} = 64$ .So, we have $\frac{4}{6}$ .
Simplify Fraction: Simplify the fraction. $\newline$ $(4) / (6)$ simplifies to $2 / 3$ . $\newline$ Therefore, $\log_{64} 16$ is equal to $2 / 3$ .