Exponential equations can often be solved by rewriting both sides to have the same base, allowing us to set the exponents equal to each other. However, when faced with an equation like \( 17 = 2^x \), where it’s not straightforward to express 17 as a power of 2, we can utilize logarithms to find the solution. This method is particularly useful when the base of the exponential expression is not easily manipulated.

To solve an exponential equation, follow these steps:

First, isolate the exponential expression. For example, in the equation \( 10^{x} + 64 = 100 \), subtract 64 from both sides to get \( 10^{x} = 36 \). This step is crucial as it sets the stage for applying logarithmic properties.

Next, determine which logarithm to use. If the base of the exponential is 10, use the common logarithm (log). If the base is not 10, use the natural logarithm (ln). In our example, since we have \( 10^{x} \), we take the common log of both sides: \( \log(10^{x}) = \log(36) \).

Then, apply the power rule of logarithms, which states that \( \log(a^b) = b \cdot \log(a) \). This allows us to bring the exponent down: \( x \cdot \log(10) = \log(36) \). Since \( \log(10) = 1 \), this simplifies to \( x = \log(36) \). This result is acceptable as is unless a numerical approximation is required.

For approximation, you can use a calculator to find \( x \approx 1.556 \) since \( \log(36) \) is a constant.

In another example, consider the equation \( 3 = 2^{x+1} \). Start by isolating the exponential expression, which is already done here. Next, since the base is not 10, take the natural log of both sides: \( \ln(3) = \ln(2^{x+1}) \). Again, apply the power rule to get \( \ln(3) = (x+1) \cdot \ln(2) \).

To solve for \( x \), divide both sides by \( \ln(2) \): \( \frac{\ln(3)}{\ln(2)} = x + 1 \). Finally, isolate \( x \) by subtracting 1: \( x = \frac{\ln(3)}{\ln(2)} - 1 \). This expression can also be approximated using a calculator, yielding \( x \approx 0.585 \).

By mastering these steps, you can confidently tackle any exponential equation, whether it involves common or natural logarithms, ensuring you can find solutions even when direct manipulation of the equation is not possible.