Chapter 3, Problem 110RE

To determine

To solve the logarithmic equation

  $\ln \sqrt{x+40}=3$

Round the result to three decimal places.

Answer to Problem 110RE

  $x\simeq 363.429$

Explanation of Solution

Given:

  $\ln \sqrt{x+40}=3$

Concept Used:

Power property of logarithmic function: $\ln u^n=n\ln u$

  $\ln u=x\iff u=e^x$

Calculation:

In order to solve the logarithmic equation

  $\ln \sqrt{x+40}=3$

First simplify it using power property of logarithmic functions.

  $\begin{array}{c}\ln \sqrt{x+40}=3\\ \frac{1}{2}\ln \left(x+40\right)=3\\ \ln \left(x+40\right)=6\end{array}$

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

As,

  $\begin{array}{c}\ln \left(x+40\right)=6\\ \Rightarrow x+40=e^6\\ \Rightarrow x+40\simeq 403.42879\\ \Rightarrow x\simeq 403.42879-40\\ \Rightarrow x\simeq 363.429\end{array}$

Thus, solution of given equation is $x\simeq 363.429$ .