Evaluate: log_(64)((1)/(4)) Answer:

Identify Properties: Identify the properties of logarithms that can be used to evaluate the expression. We have the logarithm log_(64)((1)/(4)). To evaluate this, we can use the change of base formula for logarithms, which states that log_(a)(b) = log(c)(b) / log(c)(a) for any positive base c that is not equal to 1.Apply Change of Base: Apply the change of base formula to the given logarithm. Using the change of base formula with the common base of 2 (since 64 is a power of 2 and so is 4), we get: log_(64)((1)/(4)) = log_(2)((1)/(4)) / log_(2)(64)Evaluate Logarithms: Evaluate the logarithms using the known powers of 2. We know that 2^6 = 64 and 2^(-2) = 1/4. Therefore, we can write: log_(2)((1)/(4)) = -2 (since 2 raised to the power of -2 gives 1/4) log_(2)(64) = 6 (since 2 raised to the power of 6 gives 64)Calculate Original Logarithm: Calculate the value of the original logarithm using the results from Step 3. Now we can divide the two logarithms we found: log_(64)((1)/(4)) = (-2) / 6Simplify Fraction: Simplify the fraction to get the final answer. (-2) / 6 = -1/3

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Evaluate: $\newline$ $\log _{64} \frac{1}{4}$ $\newline$ Answer:

View step-by-step help

Home

Math Problems

Precalculus

Quotient property of logarithms

Full solution

Q. Evaluate:

\newline

\log _{64} \frac{1}{4}

\newline

Answer:

Identify Properties: Identify the properties of logarithms that can be used to evaluate the expression. $\newline$ We have the logarithm $\log_{64}\left(\frac{1}{4}\right)$ . To evaluate this, we can use the change of base formula for logarithms, which states that $\log_{a}(b) = \frac{\log(c)(b)}{\log(c)(a)}$ for any positive base $c$ that is not equal to $1$ .
Apply Change of Base: Apply the change of base formula to the given logarithm. $\newline$ Using the change of base formula with the common base of $2$ (since $64$ is a power of $2$ and so is $4$ ), we get: $\newline$ $\log_{64}\left(\frac{1}{4}\right) = \frac{\log_{2}\left(\frac{1}{4}\right)}{\log_{2}(64)}$
Evaluate Logarithms: Evaluate the logarithms using the known powers of $2$ . $\newline$ We know that $2^6 = 64$ and $2^{-2} = \frac{1}{4}$ . Therefore, we can write: $\newline$ $log_{2}\left(\frac{1}{4}\right) = -2$ (since $2$ raised to the power of $-2$ gives $\frac{1}{4}$ ) $\newline$ $log_{2}(64) = 6$ (since $2$ raised to the power of $6$ gives $64$ )
Calculate Original Logarithm: Calculate the value of the original logarithm using the results from Step $3$ . $\newline$ Now we can divide the two logarithms we found: $\newline$ $\log_{64}\left(\frac{1}{4}\right) = \frac{-2}{6}$
Simplify Fraction: Simplify the fraction to get the final answer. $(-2) / 6 = -1/3$