Evaluate log base 27 of 36

Problem

(log_27)(36)

Solution

1. Apply the change of base formula to rewrite the logarithm using a common base, such as base 10 or base e

(log_27)(36)=ln(36)/ln(27)

2. Rewrite the arguments as powers of their prime factors to simplify the expression.

36=6^2

27=3^3

3. Substitute the powers back into the logarithmic expression.

(log_27)(36)=ln(6^2)/ln(3^3)

4. Apply the power rule for logarithms, which states ln(a^b)=b*ln(a)

(log_27)(36)=(2*ln(6))/(3*ln(3))

5. Expand the numerator using the product rule ln(a*b)=ln(a)+ln(b) since 6=2⋅3

(log_27)(36)=(2*(ln(2)+ln(3)))/(3*ln(3))

6. Distribute and simplify the fraction by dividing each term in the numerator by the denominator.

(log_27)(36)=(2*ln(2))/(3*ln(3))+(2*ln(3))/(3*ln(3))

(log_27)(36)=2/3*(log_3)(2)+2/3

Final Answer

(log_27)(36)=(2*(log_3)(2)+2)/3

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