Solution assignment 03 Logarithmic equations

Solve:

$\log(\displaystyle\frac{x+1}{x})=1$

The domain of the function in the left-hand side is defined by:

$\displaystyle\frac{x+1}{x}>0$

and is equal to:

$x>0$ or $x<-1$

We rewrite the equation as:

$\log(\displaystyle\frac{x+1}{x})=\log(10)$

or:

$\displaystyle\frac{x+10}{x}=10$

Cross-multiplication yields:

$x+1=10x$

$x=\displaystyle\frac{1}{9}$

This solution lies in the domain and is valid.