Math / Algebra

QuestionSolve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. $2 e^{3 x}=1294$
The solution set expressed in terms of logarithms is $\square$ \}. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the equation. Use In for natural logarithm and log for common logarithm.)
Now use a calculator to obtain a decimal approximation for the solution. The solution set is $\square$ B. (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Studdy Solution

STEP 1

1. The equation $2 e^{3x} = 1294$ is exponential.
2. We will isolate the exponential term and use logarithms to solve for $x$ .
3. We will express the solution in terms of natural logarithms (ln) or common logarithms (log).
4. We will use a calculator for decimal approximation.

STEP 2

1. Isolate the exponential term $e^{3x}$ .
2. Use logarithms to solve for $x$ .
3. Express the solution in terms of logarithms.
4. Calculate the decimal approximation.

STEP 3

First, divide both sides of the equation by 2 to isolate the exponential term:
$2 e^{3x} = 1294$ $e^{3x} = \frac{1294}{2}$ $e^{3x} = 647$

STEP 4

Take the natural logarithm (ln) of both sides to solve for $x$ . The natural logarithm is the inverse of the exponential function with base $e$ :
$\ln(e^{3x}) = \ln(647)$
Using the property of logarithms, $\ln(e^y) = y$ , we get:
$3x = \ln(647)$

STEP 5

Solve for $x$ by dividing both sides by 3:
$x = \frac{\ln(647)}{3}$
The solution set expressed in terms of natural logarithms is:
$\left\{ \frac{\ln(647)}{3} \right\}$

STEP 6

Use a calculator to find the decimal approximation for $x$ :
Calculate $\ln(647)$ and then divide by 3.
$\ln(647) \approx 6.472$ $x \approx \frac{6.472}{3} \approx 2.157$
Round to two decimal places:
$x \approx 2.16$
The solution set is:
$\{ 2.16 \}$

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Studdy Solution

STEP 1

1. The equation $2 e^{3x} = 1294$ is exponential.
2. We will isolate the exponential term and use logarithms to solve for $x$ .
3. We will express the solution in terms of natural logarithms (ln) or common logarithms (log).
4. We will use a calculator for decimal approximation.

STEP 2

1. Isolate the exponential term $e^{3x}$ .
2. Use logarithms to solve for $x$ .
3. Express the solution in terms of logarithms.
4. Calculate the decimal approximation.

STEP 3

First, divide both sides of the equation by 2 to isolate the exponential term:
$2 e^{3x} = 1294$ $e^{3x} = \frac{1294}{2}$ $e^{3x} = 647$

STEP 4

Take the natural logarithm (ln) of both sides to solve for $x$ . The natural logarithm is the inverse of the exponential function with base $e$ :
$\ln(e^{3x}) = \ln(647)$
Using the property of logarithms, $\ln(e^y) = y$ , we get:
$3x = \ln(647)$

STEP 5

Solve for $x$ by dividing both sides by 3:
$x = \frac{\ln(647)}{3}$
The solution set expressed in terms of natural logarithms is:
$\left\{ \frac{\ln(647)}{3} \right\}$

STEP 6

Use a calculator to find the decimal approximation for $x$ :
Calculate $\ln(647)$ and then divide by 3.
$\ln(647) \approx 6.472$ $x \approx \frac{6.472}{3} \approx 2.157$
Round to two decimal places:
$x \approx 2.16$
The solution set is:
$\{ 2.16 \}$

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Studdy Solution

STEP 1

1. The equation 2e3x=1294 2 e^{3x} = 1294 2e3x=1294 is exponential.2. We will isolate the exponential term and use logarithms to solve for x x x.3. We will express the solution in terms of natural logarithms (ln) or common logarithms (log).4. We will use a calculator for decimal approximation.

STEP 2

1. Isolate the exponential term e3x e^{3x} e3x.2. Use logarithms to solve for x x x.3. Express the solution in terms of logarithms.4. Calculate the decimal approximation.

STEP 3

First, divide both sides of the equation by 2 to isolate the exponential term:2e3x=1294 2 e^{3x} = 1294 2e3x=1294 e3x=12942 e^{3x} = \frac{1294}{2} e3x=21294​ e3x=647 e^{3x} = 647 e3x=647

STEP 4

STEP 5

Solve for x x x by dividing both sides by 3:x=ln⁡(647)3 x = \frac{\ln(647)}{3} x=3ln(647)​The solution set expressed in terms of natural logarithms is:{ln⁡(647)3} \left\{ \frac{\ln(647)}{3} \right\} {3ln(647)​}

STEP 6

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1. The equation $2 e^{3x} = 1294$ is exponential.
2. We will isolate the exponential term and use logarithms to solve for $x$ .
3. We will express the solution in terms of natural logarithms (ln) or common logarithms (log).
4. We will use a calculator for decimal approximation.

1. Isolate the exponential term $e^{3x}$ .
2. Use logarithms to solve for $x$ .
3. Express the solution in terms of logarithms.
4. Calculate the decimal approximation.

First, divide both sides of the equation by 2 to isolate the exponential term:
$2 e^{3x} = 1294$ $e^{3x} = \frac{1294}{2}$ $e^{3x} = 647$

Solve for $x$ by dividing both sides by 3:
$x = \frac{\ln(647)}{3}$
The solution set expressed in terms of natural logarithms is:
$\left\{ \frac{\ln(647)}{3} \right\}$