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3 \times 4 \mathrm{e}^{2-2 x}+1=4

Feb 16, 2024

3 \times 4 \mathrm{e}^{2-2 x}+1=4

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Solution by Steps

step 1

Simplify the given equation:

3 \times 4 \mathrm{e}^{2-2x} + 1 = 4

step 2

Subtract 1 from both sides to isolate the exponential term:

12\mathrm{e}^{2-2x} = 3

step 3

Divide both sides by 12 to simplify:

\mathrm{e}^{2-2x} = \frac{1}{4}

step 4

Apply the natural logarithm to both sides to solve for

x

2-2x = \ln\left(\frac{1}{4}\right)

step 5

Simplify the equation:

-2x = \ln\left(\frac{1}{4}\right) - 2

step 6

Solve for

x

x = 1 + \log(2)

Answer

x = 1 + \log(2)

Key Concept

Solving exponential equations using logarithms

Explanation

To solve an exponential equation, we isolate the exponential expression and then apply logarithms to both sides of the equation to solve for the variable. This method allows us to find the value of the variable that satisfies the original equation.