Solved on Feb 19, 2024

Solve for $x$ in the equation $5 \ln x = -15$ . Select the correct choice: A. $x = e^{-3}$ , or B. The solution is not a real number.

STEP 1

Assumptions
1. We are solving the equation for $x$ .
2. The natural logarithm function $\ln(x)$ is defined for $x > 0$ .
3. The equation is $5 \ln x = -15$ .

STEP 2

First, we need to isolate the natural logarithm term by dividing both sides of the equation by 5.
$\frac{5 \ln x}{5} = \frac{-15}{5}$

STEP 3

Simplify both sides of the equation.
$\ln x = -3$

STEP 4

To solve for $x$ , we need to rewrite the equation in exponential form, since the natural logarithm is the inverse of the exponential function with base $e$ .
$e^{\ln x} = e^{-3}$

STEP 5

Simplify the left side using the property of logarithms that states $e^{\ln x} = x$ .
$x = e^{-3}$

STEP 6

Calculate the value of $x$ using the exponential function.
$x = e^{-3}$

STEP 7

Use a calculator or a mathematical software to find the numerical value of $x$ .
$x \approx e^{-3} \approx 0.0498$

STEP 8

Round the value of $x$ to the nearest thousandth as instructed.
$x \approx 0.050$
The correct choice is A, and the value of $x$ is approximately $0.050$ .

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Solve for $x$ in the equation $5 \ln x = -15$ . Select the correct choice: A. $x = e^{-3}$ , or B. The solution is not a real number.

STEP 1

Assumptions
1. We are solving the equation for $x$ .
2. The natural logarithm function $\ln(x)$ is defined for $x > 0$ .
3. The equation is $5 \ln x = -15$ .

STEP 2

First, we need to isolate the natural logarithm term by dividing both sides of the equation by 5.
$\frac{5 \ln x}{5} = \frac{-15}{5}$

STEP 3

Simplify both sides of the equation.
$\ln x = -3$

STEP 4

To solve for $x$ , we need to rewrite the equation in exponential form, since the natural logarithm is the inverse of the exponential function with base $e$ .
$e^{\ln x} = e^{-3}$

STEP 5

Simplify the left side using the property of logarithms that states $e^{\ln x} = x$ .
$x = e^{-3}$

STEP 6

Calculate the value of $x$ using the exponential function.
$x = e^{-3}$

STEP 7

Use a calculator or a mathematical software to find the numerical value of $x$ .
$x \approx e^{-3} \approx 0.0498$

STEP 8

Round the value of $x$ to the nearest thousandth as instructed.
$x \approx 0.050$
The correct choice is A, and the value of $x$ is approximately $0.050$ .

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Solve for xxx in the equation 5ln⁡x=−155 \ln x = -155lnx=−15. Select the correct choice: A. x=e−3x = e^{-3}x=e−3, or B. The solution is not a real number.

STEP 1

Assumptions1. We are solving the equation for xxx.2. The natural logarithm function ln⁡(x)\ln(x)ln(x) is defined for x>0x > 0x>0.3. The equation is 5ln⁡x=−155 \ln x = -155lnx=−15.

STEP 2

First, we need to isolate the natural logarithm term by dividing both sides of the equation by 5.5ln⁡x5=−155\frac{5 \ln x}{5} = \frac{-15}{5}55lnx​=5−15​

STEP 3

Simplify both sides of the equation.ln⁡x=−3\ln x = -3lnx=−3

STEP 4

To solve for xxx, we need to rewrite the equation in exponential form, since the natural logarithm is the inverse of the exponential function with base eee.eln⁡x=e−3e^{\ln x} = e^{-3}elnx=e−3

STEP 5

Simplify the left side using the property of logarithms that states eln⁡x=xe^{\ln x} = xelnx=x.x=e−3x = e^{-3}x=e−3

STEP 6

Calculate the value of xxx using the exponential function.x=e−3x = e^{-3}x=e−3

STEP 7

Use a calculator or a mathematical software to find the numerical value of xxx.x≈e−3≈0.0498x \approx e^{-3} \approx 0.0498x≈e−3≈0.0498

STEP 8

Round the value of xxx to the nearest thousandth as instructed.x≈0.050x \approx 0.050x≈0.050The correct choice is A, and the value of xxx is approximately 0.0500.0500.050.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve for $x$ in the equation $5 \ln x = -15$ . Select the correct choice: A. $x = e^{-3}$ , or B. The solution is not a real number.

Assumptions
1. We are solving the equation for $x$ .
2. The natural logarithm function $\ln(x)$ is defined for $x > 0$ .
3. The equation is $5 \ln x = -15$ .

First, we need to isolate the natural logarithm term by dividing both sides of the equation by 5.
$\frac{5 \ln x}{5} = \frac{-15}{5}$

Simplify both sides of the equation.
$\ln x = -3$

To solve for $x$ , we need to rewrite the equation in exponential form, since the natural logarithm is the inverse of the exponential function with base $e$ .
$e^{\ln x} = e^{-3}$

Simplify the left side using the property of logarithms that states $e^{\ln x} = x$ .
$x = e^{-3}$

Calculate the value of $x$ using the exponential function.
$x = e^{-3}$

Use a calculator or a mathematical software to find the numerical value of $x$ .
$x \approx e^{-3} \approx 0.0498$

Round the value of $x$ to the nearest thousandth as instructed.
$x \approx 0.050$
The correct choice is A, and the value of $x$ is approximately $0.050$ .