Question

solve for x in the equation shown below.\\(\frac{1}{4}\ln(5x) + 3 = 1\\)

Explanation:

Step1: Isolate the logarithmic term

Subtract 3 from both sides of the equation:
$\frac{1}{4}\ln(5x) + 3 - 3 = 1 - 3$
$\frac{1}{4}\ln(5x) = -2$

Step2: Eliminate the coefficient of the logarithm

Multiply both sides by 4:
$4\times\frac{1}{4}\ln(5x) = -2\times4$
$\ln(5x) = -8$

Step3: Convert from logarithmic to exponential form

Recall that $\ln(a)=b$ is equivalent to $e^b = a$. So we have:
$5x = e^{-8}$

Step4: Solve for x

Divide both sides by 5:
$x = \frac{e^{-8}}{5}$
We can also write this as $x = \frac{1}{5e^{8}}$

Answer:

$x = \frac{1}{5e^{8}}$ (or $x = \frac{e^{-8}}{5}$)