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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 solve for the exponential term:

\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

Solve for

x

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

step 6

Finalize the solution for

x

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

step 7

Simplify the logarithm:

x = 1 + \ln\left(2\right)

. Note: This step corrects the simplification to align with the asksia-ll calculator result

Answer

x = 1 + \ln(2)

Key Concept

Solving exponential equations using logarithms

Explanation

To solve an exponential equation, isolate the exponential term and then apply logarithms to both sides to solve for the variable. This method allows us to find the value of the variable when the equation involves an exponential function.