Recognize Logarithm Split: Recognize that the logarithm can be split into the difference of two logarithms when dealing with a division inside the log. log((64)/(9)) = log(64) - log(9)Convert Numbers to Powers: Convert the numbers inside the logarithms to powers of their base numbers. The base of the logarithm is 10 by default, but we can express 64 as 2^6 and 9 as 3^2, which will be useful for simplification. log(64) = log(2^6) and log(9) = log(3^2)Apply Power Rule: Apply the power rule of logarithms, which states that log(a^b) = b*log(a), to both logarithms. log(2^6) = 6*log(2) and log(3^2) = 2*log(3)Substitute Expressions: Substitute the expressions from Step 3 back into the equation from Step 1. log((64)/(9)) = 6*log(2) - 2*log(3)Calculate Logarithm Values: Calculate the values of log(2) and log(3) using a calculator or logarithm table. log(2) ≈ 0.3010 and log(3) ≈ 0.4771Multiply by Coefficients: Multiply the logarithm values by their respective coefficients from Step 4. 6*log(2) ≈ 6*0.3010 = 1.8060 and 2*log(3) ≈ 2*0.4771 = 0.9542Find Final Answer: Subtract the value found for 2*log(3) from the value found for 6*log(2) to find the final answer. log((64)/(9)) ≈ 1.8060 - 0.9542 = 0.8518

Precalculus

Operations with rational exponents

Full solution

\log\left(\frac{64}{9}\right)

Recognize Logarithm Split: Recognize that the logarithm can be split into the difference of two logarithms when dealing with a division inside the log. $\log\left(\frac{64}{9}\right) = \log(64) - \log(9)$
Convert Numbers to Powers: Convert the numbers inside the logarithms to powers of their base numbers. The base of the logarithm is $10$ by default, but we can express $64$ as $2^6$ and $9$ as $3^2$ , which will be useful for simplification. $\newline$ $\log(64) = \log(2^6)$ and $\log(9) = \log(3^2)$
Apply Power Rule: Apply the power rule of logarithms, which states that $\log(a^b) = b\log(a)$ , to both logarithms. $\newline$ $\log(2^6) = 6\log(2)$ and $\log(3^2) = 2\log(3)$
Substitute Expressions: Substitute the expressions from Step $3$ back into the equation from Step $1$ . $\newline$ $\log\left(\frac{64}{9}\right) = 6\log(2) - 2\log(3)$
Calculate Logarithm Values: Calculate the values of $\log(2)$ and $\log(3)$ using a calculator or logarithm table. $\newline$ $\log(2) \approx 0.3010$ and $\log(3) \approx 0.4771$
Multiply by Coefficients: Multiply the logarithm values by their respective coefficients from Step $4$ . $\newline$ $6\log(2) \approx 6\times0.3010 = 1.8060$ and $2\log(3) \approx 2\times0.4771 = 0.9542$
Find Final Answer: Subtract the value found for $2\log(3)$ from the value found for $6\log(2)$ to find the final answer. $\newline$ $\log\left(\frac{64}{9}\right) \approx 1.8060 - 0.9542 = 0.8518$