Chapter 3, Problem 107RE

To determine

To solve the logarithmic equation

  $\ln x-\ln 5=2$

Round the result to three decimal places.

Answer to Problem 107RE

  $x\simeq 36.945$

Explanation of Solution

Given:

  $\ln x-\ln 5=2$

Concept Used:

Quotient property of logarithmic function: $\ln u-\ln v=\ln \frac{u}{v}$

  $\ln u=x\iff u=e^x$

Calculation:

In order to solve the logarithmic equation

  $\ln x-\ln 5=2$

First simplify it using quotient property of logarithmic functions.

  $\begin{array}{c}\ln x-\ln 5=2\\ \ln \left(\frac{x}{5}\right)=2\end{array}$

Now convert it into exponential equation using the property $\ln u=x\iff u=e^x$ .

As,

  $\begin{array}{c}\ln \left(\frac{x}{5}\right)=2\\ \Rightarrow \frac{x}{5}=e^2\\ \Rightarrow \frac{x}{5}\simeq 7.38905\\ \Rightarrow x\simeq 7.38905\times 5\\ \Rightarrow x\simeq 36.945\end{array}$

Thus, solution of given equation is $x\simeq 36.945$ .