₇≤ft(₇{7≤ft({7{7}})}) equals:

Correct Answer is : 1−3log⁡721-3₇21−3log72

Correct Answer is : 1−3log⁡721-3₇21−3log72

Test Index

Logarithms

Question : 45 of 85

Marks: +1, -0

\log_{7}\left(\log_{7}\sqrt{7\left(\sqrt{7\sqrt{7}}\right)}\right)

equals:

Solution:

Concept:

Simplify nested radicals using exponent laws, then evaluate the logarithms step by step.

Explanation:

Let the innermost expression be

\sqrt{7\sqrt{7}} = 7^{1/2} \cdot 7^{1/4} = 7^{3/4}

Then

\sqrt{7\sqrt{7\sqrt{7}}} = \sqrt{7 \cdot 7^{3/4}} = \sqrt{7^{7/4}} = 7^{7/8}

The inner log:

\log_7(7^{7/8}) = \frac{7}{8}

The outer log:

\log_7\left(\frac{7}{8}\right) = \log_7 7 - \log_7 8 = 1 - \log_7(2^3) = 1 - 3\log_7 2

Answer:

1 - 3\log_7 2

, which corresponds to option C.

Go to Question:

Prev Question

Next Question