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Question

Difficulty: EasyLogarithms and Change of Base

Evaluate the value of the logarithmic expression log⁡37×log⁡781\log_3 7 \times \log_7 81.

Answer: 4

Answer

The value of the expression is 4.
By applying the change of base formula log⁡781=log⁡381log⁡37\log_7 81 = \frac{\log_3 81}{\log_3 7}, the expression becomes log⁡37×log⁡381log⁡37=log⁡381\log_3 7 \times \frac{\log_3 81}{\log_3 7} = \log_3 81. Since 34=813^4 = 81, the result is 4.

Step-by-Step Solution

1
Apply the change of base chain rule log⁡ab⋅log⁡bc=log⁡ac\log_a b \cdot \log_b c = \log_a c
log⁡37×log⁡781=log⁡381\log_3 7 \times \log_7 81 = \log_3 81
By change of base, log⁡781=log⁡381log⁡37\log_7 81 = \frac{\log_3 81}{\log_3 7}, so multiplying by log⁡37\log_3 7 cancels out the common factor.
2
Evaluate log⁡381\log_3 81
4
Since 34=813^4 = 81, the logarithm base 3 of 81 is equal to 4.

Key Concept

Change of Base Property of Logarithms
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Topics

Logarithms and Change of Base

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