Question

Evaluating Logarithms Evaluate the expression.

(a)$3^{{\log }_{3}5}$ (b) $5^{{\log }_{5}27}$ (c)$e^{\ln 10}$

Short answer

1. The required value is 5.
2. The required value is27.
3. The required value is 10.

Step-by-step solution

Part a. Step 1. Given.

The given expression is $3^{{\log }_{3}5}$.

Part a. Step 2. To determine.

We have to evaluate the given expression.

Part a. Step 3. Calculation.

We’ll use the formula: $a^{{\log }_{a}b}=b$.

We compare$3^{{\log }_{3}5}$ with $a^{{\log }_{a}b}$. So, $a=3,b=5$.

Hence, $3^{{\log }_{3}5}=b=5$.

Part b. Step 1. Given.

The given exponential form is $5^{{\log }_{5}27}$.

Part b. Step 2. To determine.

We have to evaluate the given expression.

Part b. Step 3. Calculation.

We’ll use the formula: $a^{{\log }_{a}b}=b$.

We compare$5^{{\log }_{5}27}$ with $a^{{\log }_{a}b}$. So, $a=5,b=27$.

Hence, $5^{{\log }_{5}27}=b=27$.

Part c. Step 1. Given.

The given exponential form is $e^{\ln 10}$.

Part c. Step 2. To determine.

We have to evaluate the given expression.

Part c. Step 3. Calculation.

We’ll use the formula: $e^{\ln b}=b$.

We compare$e^{\ln 10}$ with $e^{\ln b}$. So, $b=10$.

Hence, $e^{\ln 10}=b=10$.