Question

(a) What are the values of \({e^{\ln 300}}\) and \(\ln \left( {{e^{300}}} \right)\)?

(b) Use your calculator to evaluate \({e^{\ln 300}}\) and \(\ln \left( {{e^{300}}} \right)\). What do you notice? Can you explain why the calculator has trouble?

Short answer

(a) The value of \({e^{\ln 300}}\) and \(\ln \left( {{e^{300}}} \right)\) is 300.

(b) \({e^{\ln 300}} = 300\) and the calculator shows an error message \(\ln \left( {{e^{300}}} \right)\) since \({e^{300}}\) is larger.

Step-by-step solution

Properties of the natural logarithm function

The properties of natural logarithm function are defined as:

1. \(\ln x = y \Leftrightarrow {e^y} = x\)
2. \(\ln \left( {{e^x}} \right) = x\,\,\,\,\,\,x \in \mathbb{R}\)
3. \({e^{\ln x}} = x\,\,\,\,\,\,\,\,\,\,\,\,\,x > 0\)
4. \(\ln e = 1\)
5. \({x^r} = {e^{r\ln x}}\)

Determine the value of \({e^{\ln 300}}\) and \(\ln \left( {{e^{300}}} \right)\)

a)

Use the properties of the natural logarithm to evaluate the value of \({e^{\ln 300}}\) and \(\ln \left( {{e^{300}}} \right)\) as shown below:

e^{In 300}=300 {∵ e^{In x}=x}

In (e³⁰⁰)=300 {∵ In e=1}

Thus, the value of \({e^{\ln 300}}\) and \(\ln \left( {{e^{300}}} \right)\) is 300.

Explain why the calculator trouble

b)

The calculator provides \({e^{\ln 300}} = 300\) and shows an error \(\ln \left( {{e^{300}}} \right)\) since \({e^{300}}\) is larger than most calculators.