Evaluate

Problem

(ln(27)+ln(9))/((log_)(27)+(log_)(9))

Solution

1. Apply the product rule for logarithms to both the numerator and the denominator to combine the terms.

ln(27)+ln(9)=ln(27⋅9)

(log_)(27)+(log_)(9)=(log_)(27⋅9)

2. Simplify the arguments of the logarithms by performing the multiplication.

27⋅9=243

3. Rewrite the expression using the simplified logarithmic terms.

ln(243)/(log_)(243)

4. Apply the change of base formula which states (log_b)(a)=ln(a)/ln(b) In this case, (log_)(243) is the common logarithm (log_10)(243) so (log_)(243)=ln(243)/ln(10)

ln(243)/ln(243)/ln(10)

5. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

ln(243)⋅ln(10)/ln(243)

6. Cancel the common factor ln(243) from the numerator and the denominator.

ln(10)

Final Answer

(ln(27)+ln(9))/((log_)(27)+(log_)(9))=ln(10)

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