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Find $\log_{4}8$

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Evaluate logarithms

Full solution

Q. Find

\log_{4}8

Identify Base: In $\log_4(8)$ , $4$ is the base. $\newline$ Rewrite $8$ as a power of $2$ because $2$ is a factor of $4$ . $\newline$ $8 = 2 \times 2 \times 2$ $\newline$ $8 = 2^3$
Express as Powers of $2$ : Now, express $4$ as a power of $2$ because $4$ is a perfect square. $4 = 2 \times 2$ $4 = 2^2$
Substitute into Logarithm: Substitute the expressions from Step $1$ and Step $2$ into the original logarithm. $\newline$ $\log_4(8)$ becomes $\log_{(2^2)}(2^3)$ .
Apply Logarithm Power Rule: Apply the logarithm power rule, which states that $\log_b(a^c) = c \cdot \log_b(a)$ . $\log_{2^2}(2^3)$ becomes $3 \cdot \log_{2^2}(2)$ .
Simplify Logarithm Bases: Since the base of the logarithm ( $2^2$ ) and the base of the argument ( $2$ ) are the same, the logarithm simplifies to the exponent of the argument. $\newline$ $\log_{2^2}(2)$ is $1$ because any number to the power of $1$ is itself. $\newline$ So, $3 \times \log_{2^2}(2)$ becomes $3 \times 1$ .
Final Calculation: Multiply to find the final answer. $3 \times 1$ equals $3$ .

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