Evaluate log base 8 of cube root of 64

Problem

(log_8)(√(3,64))

Solution

1. Simplify the expression inside the logarithm by evaluating the cube root of 64.

√(3,64)=4

2. Substitute the simplified value back into the logarithmic expression.

(log_8)(4)

3. Apply the change of base formula or express both the base and the argument as powers of the same base, which is 2.

8=2

4=2

4. Rewrite the expression using these powers.

(log_2)(2)

5. Use the property of logarithms (log_bn)(am)=m/n*(log_b)(a)

2/3*(log_2)(2)

6. Simplify the expression using the identity (log_b)(b)=1

2/3⋅1=2/3

Final Answer

(log_8)(√(3,64))=2/3

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