Expand the Logarithmic Expression

Problem

(log_27)(36)*(log_36)(49)*(log_49)(81)

Solution

1. Apply the change of base formula to each logarithmic term in the expression. The formula is (log_b)(a)=ln(a)/ln(b)

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

(log_36)(49)=ln(49)/ln(36)

(log_49)(81)=ln(81)/ln(49)

2. Multiply the fractions together and observe the cancellation of common terms in the numerators and denominators.

ln(36)/ln(27)⋅ln(49)/ln(36)⋅ln(81)/ln(49)

3. Simplify the expression by canceling ln(36) and ln(49)

ln(81)/ln(27)

4. Apply the change of base formula in reverse to combine the remaining terms into a single logarithm.

(log_27)(81)

5. Rewrite the base and the argument as powers of 3 to evaluate the logarithm.

27=3

81=3

6. Use the property (log_bn)(am)=m/n*(log_b)(a) to find the final numerical value.

(log_3)(3)=4/3*(log_3)(3)

4/3⋅1=4/3

Final Answer

(log_27)(36)*(log_36)(49)*(log_49)(81)=4/3

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