Evaluate -2 log base 8 of 5

Problem

−2*(log_8)(5)

Solution

1. Apply the power property of logarithms to move the coefficient into the exponent of the argument.

−2*(log_8)(5)=(log_8)(5(−2))

2. Simplify the exponent by calculating the reciprocal of the square.

5(−2)=1/5

5(−2)=1/25

3. Rewrite the expression with the simplified argument.

(log_8)(1/25)

4. Apply the change of base formula to express the logarithm in terms of natural logarithms or common logarithms for evaluation.

(log_8)(1/25)=ln(1/25)/ln(8)

5. Calculate the numerical value using the values ln(1/25)≈−3.2189 and ln(8)≈2.0794

(−3.2189)/2.0794≈−1.5479

Final Answer

−2*(log_8)(5)=(log_8)(1/25)≈−1.5479

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