Question

How do you solve $\ln \left( x \right)-\ln \left( x-3 \right)=\ln 5$?

Answer 1

Hint: To solve the equation we will first use the property $\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)$ Then we will apply e function on both side and simplify the equation. Then we can solve the linear equation obtained to find the value of x. Hence we have the solution of the given equation.

Complete step by step solution:
Now we are given an equation with logarithmic function. Now we know that $\ln x$ is nothing but ${{\log }_{e}}x$ . Hence we can use logarithmic properties to solve the equation.
Now we know that $\log \left( \dfrac{m}{n} \right)=\log m-\log n$
Hence using this we can write $\ln x-\ln \left( x-3 \right)$ as $\ln \left( \dfrac{x}{x-3} \right)$ .
Hence we get the given equation as $\ln \left( \dfrac{x}{x-3} \right)=\ln 5$
Now applying ${{e}^{x}}$ function on both side we get,
$\Rightarrow {{e}^{\ln \left( \dfrac{x}{x-3} \right)}}={{e}^{\ln \left( 5 \right)}}$ .
Now we know that e is the inverse function of ln. Hence we have ${{e}^{\ln x}}=x$ . Using this we get the equation as,
$\Rightarrow \dfrac{x}{x-3}=5$
Now multiplying the whole equation by x – 3 we get,
$\begin{align}
  & \Rightarrow x=5\left( x-3 \right) \\
 & \Rightarrow x=5x-15 \\
\end{align}$
Now rearranging the terms in the equation we get,
$\Rightarrow 15=5x-x$
Now the equation is a linear equation in one variable x. We know we can add and subtract the terms with the same variable of the same degree. Hence we get,
$\Rightarrow 15=4x$
Now dividing the whole equation by 4 we get,
$\Rightarrow x=\dfrac{15}{4}$

Hence the solution of the given equation is $x=\dfrac{15}{4}$ .

Note: Now note that the function is not defined on x = 3. Hence while solving such problems always check if the function is defined on such points. If the solution is such that the function is not defined on that point then the solution of the function does not exist.