Math

Question Find the solution to the equation 5ex+2=75 e^{x+2} = 7.

Studdy Solution

STEP 1

1. The equation 5ex+2=75 e^{x+2}=7 involves the natural exponential function, where ee is the base of the natural logarithm.
2. To solve for xx, we will use properties of logarithms and exponential functions.
3. The solution will provide the value of xx that satisfies the equation.

STEP 2

1. Isolate the exponential term ex+2e^{x+2}.
2. Take the natural logarithm of both sides of the equation to solve for xx.
3. Simplify the expression to find the value of xx.

STEP 3

Isolate the exponential term ex+2e^{x+2} by dividing both sides of the equation by 5.
ex+2=75 e^{x+2} = \frac{7}{5}

STEP 4

Take the natural logarithm of both sides of the equation to solve for the exponent x+2x+2.
ln(ex+2)=ln(75) \ln(e^{x+2}) = \ln\left(\frac{7}{5}\right)

STEP 5

Use the property of logarithms that states ln(ey)=y\ln(e^y) = y to simplify the left side of the equation.
x+2=ln(75) x+2 = \ln\left(\frac{7}{5}\right)

STEP 6

Subtract 2 from both sides of the equation to solve for xx.
x=ln(75)2 x = \ln\left(\frac{7}{5}\right) - 2

STEP 7

The solution to the equation is:
x=ln(75)2 x = \ln\left(\frac{7}{5}\right) - 2

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