To solve the equation [tex]\( e^{4x - 1} = 3 \)[/tex], we will use logarithmic properties. Here is the step-by-step solution:
1. Start with the given equation:
[tex]\[
e^{4x - 1} = 3
\][/tex]
2. Take the natural logarithm (ln) on both sides to break the exponential:
[tex]\[
\ln(e^{4x - 1}) = \ln(3)
\][/tex]
3. Use the property of logarithms [tex]\( \ln(e^y) = y \)[/tex] to simplify the left side:
[tex]\[
4x - 1 = \ln(3)
\][/tex]
4. Isolate [tex]\( x \)[/tex] by first adding 1 to both sides:
[tex]\[
4x = \ln(3) + 1
\][/tex]
5. Then divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\ln(3) + 1}{4}
\][/tex]
Therefore, the exact solution to the equation [tex]\( e^{4x - 1} = 3 \)[/tex] is:
[tex]\[ x = \frac{1 + \ln 3}{4} \][/tex]
So the correct answer from the given options is:
[tex]\[ x = \frac{1 + \ln 3}{4} \][/tex]
which corresponds to the option:
[tex]\[ x = \frac{1+\ln 3}{4} \][/tex]
