Start with the equation \(5^{x-1} = 5^x - 5\).

Rewrite \(5^{x-1}\) as \(\frac{5^x}{5}\), giving \(\frac{5^x}{5} = 5^x - 5\).

Multiply through by 5 to clear the fraction: \(5^x = 5 \cdot (5^x - 5)\).

This simplifies to \(5^x = 5^{x+1} - 25\).

Rearrange to get \(5^{x+1} - 5^x = 25\).

Factor out \(5^x\): \(5^x(5 - 1) = 25\).

This simplifies to \(4 \cdot 5^x = 25\).

Divide by 4: \(5^x = \frac{25}{4}\).

Take the logarithm of both sides: \(x \ln 5 = \ln \left(\frac{25}{4}\right)\).

Solve for \(x\): \(x = \frac{\ln \left(\frac{25}{4}\right)}{\ln 5}\).

Calculate \(x\) to 3 significant figures: \(x \approx 1.14\).