Math  /  Algebra

QuestionSolve the equation: 15ex+6=1215 e^{x}+6=12. x=x=\square Question Help: \square Video

Studdy Solution

STEP 1

1. The equation 15ex+6=12 15e^x + 6 = 12 is an exponential equation.
2. We will need to isolate the exponential term and use logarithms to solve for x x .

STEP 2

1. Isolate the exponential term ex e^x .
2. Apply the natural logarithm to both sides to solve for x x .
3. Simplify to find the value of x x .

STEP 3

First, subtract 6 from both sides of the equation to isolate the term with the exponential:
15ex+6=12 15e^x + 6 = 12 15ex=126 15e^x = 12 - 6 15ex=6 15e^x = 6

STEP 4

Next, divide both sides by 15 to solve for ex e^x :
15ex=6 15e^x = 6 ex=615 e^x = \frac{6}{15} ex=25 e^x = \frac{2}{5}

STEP 5

Apply the natural logarithm to both sides to solve for x x . The natural logarithm is the inverse of the exponential function:
ln(ex)=ln(25) \ln(e^x) = \ln\left(\frac{2}{5}\right)
By the property of logarithms, ln(ex)=x \ln(e^x) = x :
x=ln(25) x = \ln\left(\frac{2}{5}\right)

STEP 6

The value of x x is:
x=ln(25) x = \ln\left(\frac{2}{5}\right)
The solution is:
x=ln(25) x = \ln\left(\frac{2}{5}\right)

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