Question

(c) \log_{4} \sqrt5{256}
(f) \log_{2} \sqrt7{512}

Explanation:

Part (c)

Step1: Rewrite the radical as an exponent

The fifth root of 256 can be written as \(256^{\frac{1}{5}}\). So, \(\log_{4}\sqrt[5]{256}=\log_{4}256^{\frac{1}{5}}\).

Step2: Express 256 as a power of 4

We know that \(4^4 = 256\), so substitute 256 with \(4^4\) in the expression: \(\log_{4}(4^4)^{\frac{1}{5}}\).

Step3: Use the exponent rule \((a^m)^n=a^{mn}\)

Applying the rule, we get \(\log_{4}4^{\frac{4}{5}}\).

Step4: Use the logarithm property \(\log_{a}a^x = x\)

Here, \(a = 4\) and \(x=\frac{4}{5}\), so the value is \(\frac{4}{5}\).

Part (f)

Step1: Rewrite the radical as an exponent

The seventh root of 512 can be written as \(512^{\frac{1}{7}}\). So, \(\log_{2}\sqrt[7]{512}=\log_{2}512^{\frac{1}{7}}\).

Step2: Express 512 as a power of 2

We know that \(2^9 = 512\), so substitute 512 with \(2^9\) in the expression: \(\log_{2}(2^9)^{\frac{1}{7}}\).

Step3: Use the exponent rule \((a^m)^n=a^{mn}\)

Applying the rule, we get \(\log_{2}2^{\frac{9}{7}}\).

Step4: Use the logarithm property \(\log_{a}a^x = x\)

Here, \(a = 2\) and \(x=\frac{9}{7}\), so the value is \(\frac{9}{7}\).

Answer:

• For part (c): \(\frac{4}{5}\)
• For part (f): \(\frac{9}{7}\)