d) \( 3^{x+1}=81 \)

To solve \( 3^{x+1}=81 \), first, we recognize that \( 81 \) can be expressed as a power of \( 3 \): specifically, \( 81 = 3^4 \). This means we can rewrite the equation as \( 3^{x+1} = 3^4 \). Since the bases are the same, we can set the exponents equal to each other: \[ x + 1 = 4 \] Subtracting \( 1 \) from both sides gives \( x = 3 \). In many real-life scenarios, exponential equations come in handy, especially when predicting growth, such as in finance with compound interest or in biology with population growth. Understanding these equations lets us model and forecast trends effectively, making them invaluable across various fields and professions! A common mistake when solving exponential equations is forgetting to eliminate the base by taking logarithms. Another trick is to miscalculate the power transformation. Always double-check your base conversions and make sure the exponents are indeed equal before proceeding. These little oversights can lead to frustrating, unnecessary errors when you’re aiming for a perfect solution!