Evaluate log base 6 of 8

Problem

(log_6)(8)

Solution

1. Apply the change of base formula to rewrite the expression using common logarithms (base 10) or natural logarithms (base e.

(log_b)(a)=(log_c)(a)/(log_c)(b)

2. Substitute the values into the formula using the natural logarithm ln()

(log_6)(8)=ln(8)/ln(6)

3. Simplify the numerator by expressing 8 as 2 and applying the power rule for logarithms.

ln(8)=ln(2)=3*ln(2)

4. Simplify the denominator by expressing 6 as 2⋅3 and applying the product rule for logarithms.

ln(6)=ln(2⋅3)=ln(2)+ln(3)

5. Combine the results to find the exact value.

(log_6)(8)=(3*ln(2))/(ln(2)+ln(3))

6. Calculate the numerical value using a calculator for an approximate decimal answer.

(log_6)(8)≈2.07944/1.79176

Final Answer

(log_6)(8)≈1.1606

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