To solve the equation \( 25 \cdot 5^{x - 1} = 24 \), we can start by isolating \( 5^{x - 1} \).
-
Divide both sides by 25: \[ 5^{x - 1} = \frac{24}{25} \]
-
Taking the logarithm of both sides (we can use base 10 or natural logarithm, but we'll use natural logarithm \( \ln \) here for simplicity): \[ \ln(5^{x - 1}) = \ln\left(\frac{24}{25}\right) \]
-
Using the power rule of logarithms: \[ (x - 1) \ln(5) = \ln(24) - \ln(25) \]
-
Now solve for \( x - 1 \): \[ x - 1 = \frac{\ln(24) - \ln(25)}{\ln(5)} \]
-
Adding 1 to both sides: \[ x = \frac{\ln(24) - \ln(25)}{\ln(5)} + 1 \]
Now we can calculate the numerical value of \( x \):
Calculating \( \ln(24) \), \( \ln(25) \), and \( \ln(5) \): \[ \ln(24) \approx 3.1781 \] \[ \ln(25) \approx 3.2189 \] \[ \ln(5) \approx 1.6094 \]
Now substituting in these values: \[ x \approx \frac{3.1781 - 3.2189}{1.6094} + 1 \] \[ x \approx \frac{-0.0408}{1.6094} + 1 \] \[ x \approx -0.0253 + 1 \] \[ x \approx 0.9747 \]
Thus, the solution is approximately: \[ x \approx 0.975 \]