Chapter 3, Problem 113RE

To determine

To solve the logarithmic equation

  ${\log }_{10}\left(1-x\right)=-1$

Answer to Problem 113RE

  $x=0.9$

Explanation of Solution

Given:

  ${\log }_{10}\left(1-x\right)=-1$

Concept Used:

For $a\in R^{+}\&a\ne 1$ and $u\in R^{+}$ :

  ${\log }_{a}u=x\iff u=a^x$

  $a^{{\log }_{a}x}=x$

Calculation:

In order to solve the logarithmic equation

  ${\log }_{10}\left(1-x\right)=-1$

First convert it into exponential equation using the property ${\log }_{a}u=x\iff u=a^x$ .

As,

  $\begin{array}{c}{\log }_{10}\left(1-x\right)=-1\\ \left(1-x\right)={10}^{-1}\end{array}$

Now simplify it further to find value of x.

As,

  $\begin{array}{c}\left(1-x\right)={10}^{-1}\\ 1-x=\frac{1}{10}\\ x=1-\frac{1}{10}\\ x=\frac{9}{10}\\ x=0.9\end{array}$

Thus, solution of given equation is $x=0.9$ .