Question

Evaluate the value of the following expression:
\[ 3^{\left(2+\log_3 5\right)} \div \log_{25} 125 \]

• A. 75
• B. 40
• C. 15
• D. 60

Hint:

First apply $a^{\log_a b}=b$ to the numerator. Then write $25$ and $125$ as powers of $5$ before performing the final division.

Answer 1

The Correct Option is A

Step 1: Simplify the exponent in the numerator.
\[ 3^{\left(2+\log_3 5\right)} = 3^2 \cdot 3^{\log_3 5} \]
Using the identity \(a^{\log_a b} = b\), we get:
\[ 3^2 \cdot 5 = 9 \cdot 5 = 45 \] Step 2: Simplify the denominator.
\[ \log_{25} 125 \]
Write both numbers in terms of base \(5\):
\[ 25 = 5^2,\quad 125 = 5^3 \]
\[ \log_{25} 125 = \frac{\log 5^3}{\log 5^2} = \frac{3}{2} \] Step 3: Divide the results.
\[ \frac{45}{\frac{3}{2}} = 45 \times \frac{2}{3} = 30 \] Final Answer:
\[ \boxed{30} \]

Answer 2

Concept:

• Use the inverse relationship between exponents and logarithms.
• Convert both numbers in the denominator to powers of the same base.

Step 1: Evaluate the exponential part.
$3^{2+\log_3 5}=3^2\cdot3^{\log_3 5}=9\cdot5=45$.

Step 2: Evaluate the logarithm in the divisor.
Since $25=5^2$ and $125=5^3$, $\log_{25}125=\dfrac{3}{2}$.

Step 3: Divide by the fractional value.
$45\div\dfrac{3}{2}=45\times\dfrac{2}{3}=30$.

Final Answer: $30$