Identify base and argument: Identify the base of the logarithm and the argument. The base of the logarithm is 64, and the argument is 1/4.Express argument as power of base: Express the argument 1/4 as a power of the base 64. Since 64 is 2 raised to the 6th power (64 = 2^6), we need to express 1/4 in terms of a power of 2. We know that 1/4 is 2 raised to the power of -2 (1/4 = 2^-2).Express power of base as power of 64: Express 2^-2 as a power of 64. Since 64 is 2^6, we can write 2^-2 as (2^6)^(-1/3) because (2^6)^(-1/3) = 2^(6*(-1/3)) = 2^-2.Write logarithm with new expression: Write the logarithm with the new expression for 1/4. log_(64)((2^6)^(-1/3))Apply power property of logarithms: Apply the power property of logarithms. The power property of logarithms states that log_b(a^c) = c * log_b(a). Therefore, we have: log_(64)((2^6)^(-1/3)) = (-1/3) * log_(64)(2^6)Evaluate logarithm: Evaluate log_(64)(2^6). Since the base of the logarithm (64) is equal to 2^6, log_(64)(2^6) = 1.Multiply result by exponent: Multiply the result from Step 6 by the exponent from Step 5. (-1/3) * 1 = -1/3Conclude final answer: Conclude the final answer. The value of log_(64)(1/4) is -1/3.

Resources

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

View step-by-step help

Home

Math Problems

Algebra 2

Evaluate logarithms using properties

Full solution

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

Identify base and argument: Identify the base of the logarithm and the argument. $\newline$ The base of the logarithm is $64$ , and the argument is $\frac{1}{4}$ .
Express argument as power of base: Express the argument $\frac{1}{4}$ as a power of the base $64$ . $\newline$ Since $64$ is $2$ raised to the $6$ th power ( $64 = 2^6$ ), we need to express $\frac{1}{4}$ in terms of a power of $2$ . We know that $\frac{1}{4}$ is $2$ raised to the power of $64$ $0$ ( $64$ $1$ ).
Express power of base as power of $64$ : Express $2^{-2}$ as a power of $64$ . $\newline$ Since $64$ is $2^6$ , we can write $2^{-2}$ as $(2^6)^{-\frac{1}{3}}$ because $(2^6)^{-\frac{1}{3}} = 2^{6(-\frac{1}{3})} = 2^{-2}$ .
Write logarithm with new expression: Write the logarithm with the new expression for $\frac{1}{4}$ . $\newline$ $\log_{64}\left((2^6)^{-\frac{1}{3}}\right)$
Apply power property of logarithms: Apply the power property of logarithms. $\newline$ The power property of logarithms states that $\log_b(a^c) = c \cdot \log_b(a)$ . Therefore, we have: $\newline$ $\log_{64}((2^6)^{-\frac{1}{3}}) = -\frac{1}{3} \cdot \log_{64}(2^6)$
Evaluate logarithm: Evaluate $\log_{64}(2^6)$ . $\newline$ Since the base of the logarithm $(64)$ is equal to $2^6$ , $\log_{64}(2^6) = 1$ .
Multiply result by exponent: Multiply the result from Step $6$ by the exponent from Step $5$ . $\newline$ $(-\frac{1}{3}) \times 1 = -\frac{1}{3}$
Conclude final answer: Conclude the final answer. $\newline$ The value of $\log_{64}(\frac{1}{4})$ is $-\frac{1}{3}$ .