The given equation is:

$2^{5x - 1} = 3$

Steps to solve:

1. Take the natural logarithm (ln) on both sides of the equation to help simplify the exponent: $\ln(2^{5x - 1}) = \ln(3)$

2. Apply the logarithmic property $\ln(a^b) = b \ln(a)$ to move the exponent down: $(5x - 1) \ln(2) = \ln(3)$

3. Solve for $x$:

   First, isolate $5x - 1$: $5x - 1 = \frac{\ln(3)}{\ln(2)}$

   Now, add 1 to both sides: $5x = \frac{\ln(3)}{\ln(2)} + 1$

4. Solve for $x$: $x = \frac{\frac{\ln(3)}{\ln(2)} + 1}{5}$

Now, let's calculate the values of $\ln(3)$ and $\ln(2)$ to get the final answer.

$\ln(3) \approx 1.0986, \quad \ln(2) \approx 0.6931$

Thus,

$x = \frac{\frac{1.0986}{0.6931} + 1}{5}$

$x = \frac{1.585 + 1}{5}$

$x = \frac{2.585}{5} \approx 0.5170$

Final Answer:

The solution is approximately: $x \approx 0.5170$

Would you like further explanation or assistance with another problem?

Here are some related questions to explore:

1. How can you use logarithms to solve exponential equations?
2. What is the inverse of the exponential function $2^x$?
3. How do you solve equations with logarithms as the base?
4. What is the natural logarithm and why is it often used in solving equations like these?
5. How do you graph the function $2^{5x - 1}$ and find its intercept?

Tip: When solving equations involving exponents, logarithms are often the easiest way to "bring down" the exponent and isolate the variable.