91Ó°ÊÓ

To solve an exponential equation, the first crucial step is to isolate the exponential term. In mathematical problems involving exponents, the term containing the exponent often determines the rest of the calculation. In this exercise, we start with the equation

• \( 4 + 3^{5x} = 8 \).

The goal is to simplify it by removing the constant term from the left side. By doing this, we can work directly with the term containing the variable.
We subtract 4 from both sides, resulting in

• \( 3^{5x} = 4 \).

This simplification allows us to focus entirely on the exponential part, making it ready for more advanced operations, like the application of logarithms.

Logarithms are handy tools when dealing with exponents. Once you've isolated the exponential term, it's useful to apply logarithms to both sides of the equation. For our example, we apply the natural logarithm (ln) to

• \( 3^{5x} = 4 \).

By using the logarithm property, \( \ln(a^b) = b \cdot \ln(a) \), the exponent can be "brought down" to become a coefficient. This transformation turns our equation into

• \( 5x \cdot \ln(3) = \ln(4) \).

The properties of logarithms are powerful because they allow multiplication inside the logarithm to become multiplication outside, making it easier to isolate and solve for variables like \( x \). This step simplifies the equation significantly, preparing it for straightforward algebraic manipulation.

Now that we have transformed the exponential equation using logarithms, solving for \( x \) is a straightforward task. Our previous result is

• \( 5x \cdot \ln(3) = \ln(4) \).

To isolate \( x \), we divide both sides by \( 5 \ln(3) \), giving us

• \( x = \frac{\ln(4)}{5 \ln(3)} \).

This formula now can be computed using a calculator. Calculating the logarithms and dividing gives an approximate value for \( x \). Here, we find

• \( \ln(4) \approx 1.3863 \) and \( \ln(3) \approx 1.0986 \).

Substituting these into the equation, we get

• \( x \approx \frac{1.3863}{5 \times 1.0986} \approx 0.2521 \).

Always remember to round your final answer appropriately, ensuring it matches the precision required by the exercise.