Question Find the value of $x$ in the equation $e^{x}=5$ . The correct statements are: $x=\ln 5$ , $x=\log _{5} e$ , and to the nearest hundredth, $x \approx 1.61$ .

Studdy Solution

STEP 1

1. The expression $e^{x}=5$ involves the natural exponential function, where $e$ is the base of the natural logarithm.
2. The natural logarithm function, denoted as $\ln$ , is the inverse of the natural exponential function.
3. The logarithm function can be used to solve for $x$ in the equation $e^{x}=5$ .
4. The logarithm base conversion formula can be used to convert between different bases of logarithms.

STEP 2

1. Solve the equation $e^{x}=5$ for $x$ .
2. Evaluate the natural logarithm of $5$ to find the value of $x$ .
3. Determine if $x=\log _{5} e$ is true by using the base conversion formula for logarithms.
4. Calculate the value of $x$ to the nearest hundredth.
5. Assess the truth of the statement $x=\frac{5}{e}$ .

STEP 3

Take the natural logarithm of both sides of the equation to solve for $x$ .
$\ln(e^{x}) = \ln(5)$

STEP 4

Use the property of logarithms that $\ln(e^{x}) = x$ to simplify the left side of the equation.
$x = \ln(5)$

STEP 5

Evaluate $\ln(5)$ to find the value of $x$ .
$x \approx \ln(5)$

STEP 6

Use a calculator or other computational tool to find the numerical value of $\ln(5)$ to the nearest hundredth.
$x \approx 1.61$

STEP 7

Use the base conversion formula for logarithms to determine if $x=\log _{5} e$ is true.
$\log _{5} e = \frac{\ln(e)}{\ln(5)}$

STEP 8

Simplify the right side of the equation, knowing that $\ln(e) = 1$ .
$\log _{5} e = \frac{1}{\ln(5)}$

STEP 9

Evaluate the right side of the equation using the previously found value of $\ln(5)$ .
$\log _{5} e \approx \frac{1}{1.61}$

STEP 10

Calculate the numerical value to the nearest hundredth.
$\log _{5} e \approx 0.62$

STEP 11

Compare the calculated value of $x$ to the given approximations to determine which are correct.
$x \approx 1.61$

STEP 12

Assess the truth of the statement $x=\frac{5}{e}$ by comparing it to the previously found value of $x$ .
$x \neq \frac{5}{e}$
Based on the steps above, the following statements about the value of $x$ are true: - $x=\ln 5$ - To the nearest hundredth, $x \approx 1.61$
The other statements are false: - To the nearest hundredth, $x \approx 0.70$ is false because $x \approx 1.61$ . - $x=\log _{5} e$ is false because $\log _{5} e \approx 0.62$ . - To the nearest hundredth, $x \approx 1.84$ is false because $x \approx 1.61$ . - $x=\frac{5}{e}$ is false because $x$ is the exponent to which $e$ must be raised to get 5, not a fraction.

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