To solve the equation \( 25 \cdot 5^{x-1} = 24 \), we can start by rewriting \( 25 \) as \( 5^2 \):
\[ 5^2 \cdot 5^{x-1} = 24 \]
Next, we can use the properties of exponents to combine the terms on the left side:
\[ 5^{2 + (x - 1)} = 24 \]
This simplifies to:
\[ 5^{x + 1} = 24 \]
Now, we can take the logarithm of both sides to solve for \( x \). We will use the natural logarithm, but you could use any logarithm:
\[ \ln(5^{x + 1}) = \ln(24) \]
Using the power rule of logarithms, we can rewrite the left side:
\[ (x + 1) \ln(5) = \ln(24) \]
Now, we can solve for \( x + 1 \):
\[ x + 1 = \frac{\ln(24)}{\ln(5)} \]
Subtracting 1 from both sides gives:
\[ x = \frac{\ln(24)}{\ln(5)} - 1 \]
To get a numerical answer, we can calculate the values:
\[ \ln(24) \approx 3.178 \] \[ \ln(5) \approx 1.609 \]
Now plug these values in:
\[ x \approx \frac{3.178}{1.609} - 1 \approx 1.975 - 1 \approx 0.975 \]
Thus, the solution is approximately:
\[ \boxed{0.975} \]