The given equation is:

$4^x = 26$

Solving for $x$:

1. Take the natural logarithm (ln) on both sides:

   $\ln(4^x) = \ln(26)$

2. Use the logarithm property: $\ln(a^b) = b \ln(a)$:

   $x \ln(4) = \ln(26)$

3. Solve for $x$:

   $x = \frac{\ln(26)}{\ln(4)}$

4. Compute the values:

   $x = \frac{\ln(26)}{\ln(4)} = \frac{3.2581}{1.3863} \approx 2.3505$

Final Answer:

$x \approx 2.3505$

(Rounded to four decimal places as required.)

Would you like a more detailed explanation or have any questions?

Here are five related questions you might find helpful:

1. How do logarithm properties help in solving exponential equations?
2. What is the difference between natural logarithm ($\ln$) and common logarithm ($\log$)?
3. How can logarithms be used to solve equations involving exponents?
4. What is the change of base formula for logarithms?
5. Can we solve the equation $4^x = 26$ using logarithms with base 4 instead?

Tip: When solving exponential equations, taking the logarithm on both sides helps bring the exponent down for easier calculations.