Math / Algebra

QuestionFind the solution of the exponential equation $20 e^{x}-12=14$ The exact solution, in terms of the natural logarithm is: $x=$ $\square$ The approximate solution, accurate to 4 decimal places is: $x=$ $\square$

Studdy Solution

STEP 1

1. The equation $20e^x - 12 = 14$ is exponential.
2. We will need to isolate the exponential term and use logarithms to solve for $x$ .

STEP 2

1. Isolate the exponential term $e^x$ .
2. Use the natural logarithm to solve for $x$ .
3. Calculate the exact solution.
4. Calculate the approximate solution.

STEP 3

First, add 12 to both sides of the equation to isolate the term with $e^x$ :
$20e^x - 12 = 14$ $20e^x = 14 + 12$ $20e^x = 26$

STEP 4

Next, divide both sides by 20 to solve for $e^x$ :
$e^x = \frac{26}{20}$ $e^x = \frac{13}{10}$

STEP 5

Take the natural logarithm of both sides to solve for $x$ :
$\ln(e^x) = \ln\left(\frac{13}{10}\right)$ $x = \ln\left(\frac{13}{10}\right)$
This is the exact solution in terms of the natural logarithm.

STEP 6

Calculate the approximate solution by evaluating the natural logarithm:
$x \approx \ln\left(\frac{13}{10}\right) \approx 0.2624$
The exact solution is:
$x = \ln\left(\frac{13}{10}\right)$
The approximate solution, accurate to 4 decimal places, is:
$x \approx 0.2624$

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QuestionFind the solution of the exponential equation $20 e^{x}-12=14$ The exact solution, in terms of the natural logarithm is: $x=$ $\square$ The approximate solution, accurate to 4 decimal places is: $x=$ $\square$

Studdy Solution

STEP 1

1. The equation $20e^x - 12 = 14$ is exponential.
2. We will need to isolate the exponential term and use logarithms to solve for $x$ .

STEP 2

1. Isolate the exponential term $e^x$ .
2. Use the natural logarithm to solve for $x$ .
3. Calculate the exact solution.
4. Calculate the approximate solution.

STEP 3

First, add 12 to both sides of the equation to isolate the term with $e^x$ :
$20e^x - 12 = 14$ $20e^x = 14 + 12$ $20e^x = 26$

STEP 4

Next, divide both sides by 20 to solve for $e^x$ :
$e^x = \frac{26}{20}$ $e^x = \frac{13}{10}$

STEP 5

Take the natural logarithm of both sides to solve for $x$ :
$\ln(e^x) = \ln\left(\frac{13}{10}\right)$ $x = \ln\left(\frac{13}{10}\right)$
This is the exact solution in terms of the natural logarithm.

STEP 6

Calculate the approximate solution by evaluating the natural logarithm:
$x \approx \ln\left(\frac{13}{10}\right) \approx 0.2624$
The exact solution is:
$x = \ln\left(\frac{13}{10}\right)$
The approximate solution, accurate to 4 decimal places, is:
$x \approx 0.2624$

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QuestionFind the solution of the exponential equation 20ex−12=1420 e^{x}-12=1420ex−12=14 The exact solution, in terms of the natural logarithm is: x=x=x= □\square□ The approximate solution, accurate to 4 decimal places is: x=x=x= □\square□

Studdy Solution

STEP 1

1. The equation 20ex−12=14 20e^x - 12 = 14 20ex−12=14 is exponential.2. We will need to isolate the exponential term and use logarithms to solve for x x x.

STEP 2

1. Isolate the exponential term ex e^x ex.2. Use the natural logarithm to solve for x x x.3. Calculate the exact solution.4. Calculate the approximate solution.

STEP 3

First, add 12 to both sides of the equation to isolate the term with ex e^x ex:20ex−12=14 20e^x - 12 = 14 20ex−12=14 20ex=14+12 20e^x = 14 + 12 20ex=14+12 20ex=26 20e^x = 26 20ex=26

STEP 4

Next, divide both sides by 20 to solve for ex e^x ex:ex=2620 e^x = \frac{26}{20} ex=2026​ ex=1310 e^x = \frac{13}{10} ex=1013​

STEP 5

Take the natural logarithm of both sides to solve for x x x:ln⁡(ex)=ln⁡(1310) \ln(e^x) = \ln\left(\frac{13}{10}\right) ln(ex)=ln(1013​) x=ln⁡(1310) x = \ln\left(\frac{13}{10}\right) x=ln(1013​)This is the exact solution in terms of the natural logarithm.

STEP 6

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QuestionFind the solution of the exponential equation $20 e^{x}-12=14$ The exact solution, in terms of the natural logarithm is: $x=$ $\square$ The approximate solution, accurate to 4 decimal places is: $x=$ $\square$

1. The equation $20e^x - 12 = 14$ is exponential.
2. We will need to isolate the exponential term and use logarithms to solve for $x$ .

1. Isolate the exponential term $e^x$ .
2. Use the natural logarithm to solve for $x$ .
3. Calculate the exact solution.
4. Calculate the approximate solution.

First, add 12 to both sides of the equation to isolate the term with $e^x$ :
$20e^x - 12 = 14$ $20e^x = 14 + 12$ $20e^x = 26$

Next, divide both sides by 20 to solve for $e^x$ :
$e^x = \frac{26}{20}$ $e^x = \frac{13}{10}$

Take the natural logarithm of both sides to solve for $x$ :
$\ln(e^x) = \ln\left(\frac{13}{10}\right)$ $x = \ln\left(\frac{13}{10}\right)$
This is the exact solution in terms of the natural logarithm.