Question

Evaluate \[ 9\left(\log_{27}81+\log_{9}81+\log_{3}243\right). \]

• A. 25
• B. 35
• C. 55
• D. 75

Hint:

Remember: \[ \log_{a^m}(a^n)=\frac{n}{m}. \] Convert all numbers to the same base before evaluating logarithms.

Answer 1

The Correct Option is D

Concept: Express each number as a power of the same base and use the property \[ \boxed{\log_{a^m}(a^n)=\frac{n}{m}.} \]

Step 1: Evaluate each logarithm.
Since \[ 27=3^3,\qquad 81=3^4,\qquad 243=3^5, \] we have \[ \log_{27}81 =\log_{3^3}(3^4) =\frac{4}{3}, \] \[ \log_{9}81 =\log_{3^2}(3^4) =\frac{4}{2} =2, \] \[ \log_{3}243 =\log_{3}(3^5) =5. \]

Step 2: Add the logarithms.
\[ \frac{4}{3}+2+5 =\frac{4}{3}+7 =\frac{25}{3}. \]

Step 3: Multiply by 9.
\[ 9\times\frac{25}{3} =3\times25 =75. \]

Step 4: Final conclusion.
Hence, \[ 9\left(\log_{27}81+\log_{9}81+\log_{3}243\right) =\boxed{75}. \]