QuestionSolve the following exponential equation. Express the solution set in terms of natural logarithms or common los $e^{5 x-2}-7=25150$

Studdy Solution

STEP 1

1. The equation $e^{5x-2} - 7 = 25150$ is an exponential equation.
2. We will use natural logarithms to solve for $x$ .

STEP 2

1. Isolate the exponential term $e^{5x-2}$ .
2. Apply the natural logarithm to both sides.
3. Solve for $x$ .

STEP 3

Add 7 to both sides of the equation to isolate the exponential term:
$e^{5x-2} - 7 = 25150$ $e^{5x-2} = 25150 + 7$ $e^{5x-2} = 25157$

STEP 4

Apply the natural logarithm to both sides of the equation to remove the exponential:
$\ln(e^{5x-2}) = \ln(25157)$
Using the property of logarithms, $\ln(e^y) = y$ , we simplify the left side:
$5x - 2 = \ln(25157)$

STEP 5

Solve for $x$ by first adding 2 to both sides:
$5x - 2 = \ln(25157)$ $5x = \ln(25157) + 2$
Then divide both sides by 5:
$x = \frac{\ln(25157) + 2}{5}$
The solution set in terms of natural logarithms is:
$x = \frac{\ln(25157) + 2}{5}$

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