2. \( 25^{x-5}=5 \)

To solve the equation \( 25^{x-5} = 5 \), start by expressing 25 as a power of 5, since \( 25 = 5^2 \). This gives us \( (5^2)^{x-5} = 5 \), which simplifies to \( 5^{2(x-5)} = 5^1 \). Now, because the bases are the same, we can set the exponents equal to each other: \[ 2(x - 5) = 1 \] Next, solve for \( x \) by first dividing both sides by 2: \[ x - 5 = \frac{1}{2} \] Finally, add 5 to both sides: \[ x = 5 + \frac{1}{2} = 5.5 \] So, \( x = 5.5 \). --- That's how you crack a logarithmic equation! But remember, it's always good fun to try graphing both sides to see where they intersect – it gives a visual sense of your solution! Also, pay attention to details like base conversions; they can make or break your equation-solving flow. Keep practicing!