To solve the equation \( 5 \cdot 11^{4x} = 40 \), we can start by isolating \( 11^{4x} \).
-
Divide both sides by 5: \[ 11^{4x} = \frac{40}{5} \] \[ 11^{4x} = 8 \]
-
Take the logarithm of both sides (we can use natural logarithm or common logarithm; here we'll use natural logarithm): \[ \ln(11^{4x}) = \ln(8) \]
-
Use the property of logarithms (\(\ln(a^b) = b \cdot \ln(a)\)): \[ 4x \cdot \ln(11) = \ln(8) \]
-
Solve for \(x\): \[ x = \frac{\ln(8)}{4 \ln(11)} \]
-
Calculate the values:
- First, find the natural logarithms: \[ \ln(8) \approx 2.0794 \] \[ \ln(11) \approx 2.3979 \]
- Now plug these values into the equation: \[ x = \frac{2.0794}{4 \cdot 2.3979} = \frac{2.0794}{9.5916} \approx 0.2175 \]
-
Approximate to two decimal places: \[ x \approx 0.22 \]
Therefore, the solution is: \[ \boxed{0.22} \]