QuestionSolve the equation: $e^{3 x} = 5$

Studdy Solution

STEP 1

Assumptions1. The equation is in the form of $e^{3x} =5$ . We are solving for the variable $x$
3. The base of the natural logarithm is $e$

STEP 2

To solve for $x$ , we need to isolate $x$ on one side of the equation. We can do this by taking the natural logarithm of both sides of the equation.
$\ln(e^{x}) = \ln(5)$

STEP 3

By properties of logarithms, we know that $\ln(e^{3x})$ can be simplified to $3x \cdot \ln(e)$ .
$3x \cdot \ln(e) = \ln(5)$

STEP 4

Since the natural logarithm of $e$ is1, we can simplify the equation further.
$3x = \ln()$

STEP 5

Finally, to solve for $x$ , we divide both sides of the equation by3.
$x = \frac{\ln(5)}{3}$

STEP 6

Now, we can calculate the value of $x$ .
$x = \frac{\ln(5)}{3} \approx0.54$ So, the solution to the equation $e^{3x} =5$ is $x \approx0.54$ .

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QuestionSolve the equation: $e^{3 x} = 5$

Studdy Solution

STEP 1

Assumptions1. The equation is in the form of $e^{3x} =5$ . We are solving for the variable $x$
3. The base of the natural logarithm is $e$

STEP 2

To solve for $x$ , we need to isolate $x$ on one side of the equation. We can do this by taking the natural logarithm of both sides of the equation.
$\ln(e^{x}) = \ln(5)$

STEP 3

By properties of logarithms, we know that $\ln(e^{3x})$ can be simplified to $3x \cdot \ln(e)$ .
$3x \cdot \ln(e) = \ln(5)$

STEP 4

Since the natural logarithm of $e$ is1, we can simplify the equation further.
$3x = \ln()$

STEP 5

Finally, to solve for $x$ , we divide both sides of the equation by3.
$x = \frac{\ln(5)}{3}$

STEP 6

Now, we can calculate the value of $x$ .
$x = \frac{\ln(5)}{3} \approx0.54$ So, the solution to the equation $e^{3x} =5$ is $x \approx0.54$ .

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QuestionSolve the equation: e3x=5e^{3 x} = 5e3x=5

Studdy Solution

STEP 1

Assumptions1. The equation is in the form of e3x=5e^{3x} =5e3x=5 . We are solving for the variable xxx3. The base of the natural logarithm is eee

STEP 2

To solve for xxx, we need to isolate xxx on one side of the equation. We can do this by taking the natural logarithm of both sides of the equation.ln⁡(ex)=ln⁡(5)\ln(e^{x}) = \ln(5)ln(ex)=ln(5)

STEP 3

By properties of logarithms, we know that ln⁡(e3x)\ln(e^{3x})ln(e3x) can be simplified to 3x⋅ln⁡(e)3x \cdot \ln(e)3x⋅ln(e).3x⋅ln⁡(e)=ln⁡(5)3x \cdot \ln(e) = \ln(5)3x⋅ln(e)=ln(5)

STEP 4

Since the natural logarithm of eee is1, we can simplify the equation further.3x=ln⁡()3x = \ln()3x=ln()

STEP 5

Finally, to solve for xxx, we divide both sides of the equation by3.x=ln⁡(5)3x = \frac{\ln(5)}{3}x=3ln(5)​

STEP 6

Now, we can calculate the value of xxx.x=ln⁡(5)3≈0.54x = \frac{\ln(5)}{3} \approx0.54x=3ln(5)​≈0.54So, the solution to the equation e3x=5e^{3x} =5e3x=5 is x≈0.54x \approx0.54x≈0.54.

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QuestionSolve the equation: $e^{3 x} = 5$

Assumptions1. The equation is in the form of $e^{3x} =5$ . We are solving for the variable $x$
3. The base of the natural logarithm is $e$

To solve for $x$ , we need to isolate $x$ on one side of the equation. We can do this by taking the natural logarithm of both sides of the equation.
$\ln(e^{x}) = \ln(5)$

By properties of logarithms, we know that $\ln(e^{3x})$ can be simplified to $3x \cdot \ln(e)$ .
$3x \cdot \ln(e) = \ln(5)$

Since the natural logarithm of $e$ is1, we can simplify the equation further.
$3x = \ln()$

Finally, to solve for $x$ , we divide both sides of the equation by3.
$x = \frac{\ln(5)}{3}$

Now, we can calculate the value of $x$ .
$x = \frac{\ln(5)}{3} \approx0.54$ So, the solution to the equation $e^{3x} =5$ is $x \approx0.54$ .