Question Simplify the expressions: (a) $e^{\ln (\sqrt{7})}$ , (b) $e^{\ln (1 / \pi)}$ , (c) $10^{\log (15)}$ .

Studdy Solution

STEP 1

Assumptions
1. We are dealing with exponential and logarithmic functions.
2. The base of the natural logarithm $\ln$ is $e$ .
3. The properties of logarithms and exponents will be used to simplify the expressions.
4. The base of the common logarithm $\log$ is $10$ .

STEP 2

Evaluate the expression $e^{\ln (\sqrt{7})}$ .
We will use the property that $e^{\ln x} = x$ for any positive number $x$ .

STEP 3

Recognize that $\sqrt{7}$ is a positive number, so we can apply the property directly.
$e^{\ln (\sqrt{7})} = \sqrt{7}$

STEP 4

The expression simplifies to:
$e^{\ln (\sqrt{7})} = \sqrt{7}$

STEP 5

Evaluate the expression $e^{\ln (1 / \pi)}$ .
Again, we will use the property that $e^{\ln x} = x$ for any positive number $x$ .

STEP 6

Recognize that $1 / \pi$ is a positive number, so we can apply the property directly.
$e^{\ln (1 / \pi)} = \frac{1}{\pi}$

STEP 7

The expression simplifies to:
$e^{\ln (1 / \pi)} = \frac{1}{\pi}$

STEP 8

Evaluate the expression $10^{\log (15)}$ .
We will use the property that $10^{\log x} = x$ for any positive number $x$ .

STEP 9

Recognize that $15$ is a positive number, so we can apply the property directly.
$10^{\log (15)} = 15$

STEP 10

The expression simplifies to:
$10^{\log (15)} = 15$
Solutions: (a) $e^{\ln (\sqrt{7})} = \sqrt{7}$ (b) $e^{\ln (1 / \pi)} = \frac{1}{\pi}$ (c) $10^{\log (15)} = 15$

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