Question

e^{\ln 1} = \quad e^{\ln 5x} =

Explanation:

Step1: Recall the inverse property of exponential and logarithmic functions

The exponential function \( e^x \) and the natural logarithmic function \( \ln x \) (which is \( \log_e x \)) are inverse functions of each other. So, for any positive real number \( a \), we have the property \( e^{\ln a}=a \).

Step2: Solve \( e^{\ln 1} \)

Using the property \( e^{\ln a}=a \), when \( a = 1 \), we substitute into the formula. So \( e^{\ln 1}=1 \) because \( \ln 1 = 0 \) and \( e^0=1 \), but also directly from the inverse property \( e^{\ln a}=a \) with \( a = 1 \).

Step3: Solve \( e^{\ln 5x} \)

Again, using the inverse property of exponential and natural logarithm \( e^{\ln a}=a \), here \( a = 5x \) (assuming \( 5x>0 \) or \( x > 0 \) since the domain of \( \ln \) function requires the argument to be positive). So \( e^{\ln 5x}=5x \).

Answer:

For \( e^{\ln 1} \), the answer is \( 1 \); for \( e^{\ln 5x} \), the answer is \( 5x \)