Question

evaluate the expression without using a calculator.

in ( e ^ { 6 } )

( ln e ^ { 6 } = )

Explanation:

Step1: Use the logarithm power rule

The logarithm power rule is \(\ln a^b = b\ln a\). For the expression \(\ln e^{6}\), here \(a = e\) and \(b = 6\). So, \(\ln e^{6}=6\ln e\).

Step2: Use the property of natural logarithm

We know that the natural logarithm \(\ln x\) has the property \(\ln e = 1\) (since \(y=\ln x\) is the inverse function of \(y = e^{x}\), and \(e^{1}=e\)). Substituting \(\ln e = 1\) into \(6\ln e\), we get \(6\times1\).

Answer:

\(6\)