Question

In the following exercises, solve for $x$.

$\log x+\log (x-15)=2$

Short answer

By solving the equation $\log x+\log (x-15)=2$, the value of $x$is $20$

Step-by-step solution

Step 1. Given

An expression $\log x+\log (x-15)=2$

To solve the expression for $x$

Step 2. Use the Product property

By Product Property of logarithm,

${\log }_{a}M+{\log }_{a}N={\log }_{a}MN$

$\log \left(x\right(x-15\left)\right)=2$

Step 3. Rewrite in exponential form

$x(x-15)={10}^2$

$x^2-15x=100$

$x^2-15x-100=0$

Solve the equation.

$$\begin{gathered} x^2-20x+5x-100=0 \\ (x-20)(x+5)=0 \\ x=-5,20 \end{gathered}$$

Logarithm for a negative number does not exist.

So, $x=20$

Step 4. Check the solution

Substitute $x=20$in the original equation and solve.

$$\begin{gathered} \log \left(20\right)+\log (20-15)=2 \\ \log \left(100\right)=2 \end{gathered}$$

$2=2$

Hence the equation has been checked.