Expand \( \log_2 \frac{x^5}{8} \) Using Logarithmic Properties - Math Solver AI

$Math Problem$

Mathematics High School

Question Content

Expand the logarithm and simplify if possible: \( \log_2 \frac{x^5}{8} \)

Correct Answer

C

Detailed Solution Steps

1
Apply the Quotient Rule: \( \log_2 \frac{x^5}{8} = \log_2 x^5 - \log_2 8 \)
2
Apply the Power Rule to \( \log_2 x^5 \): \( 5 \log_2 x \)
3
Simplify \( \log_2 8 \): \( \log_2 2^3 = 3 \) (since \( \log_b b^k = k \))
4
Combine results: \( 5 \log_2 x - 3 = -3 + 5 \log_2 x \)

Knowledge Points Involved

1

Quotient Rule for Logarithms

For \( b > 0, b \neq 1, M, N > 0 \), \( \log_b \frac{M}{N} = \log_b M - \log_b N \).

2

Power Rule for Logarithms

For \( b > 0, b \neq 1, M > 0, n \) real, \( \log_b M^n = n \log_b M \).

3

Logarithm of a Power

For \( b > 0, b \neq 1, k \) real, \( \log_b b^k = k \) (inverse property of logarithms).

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