Question

In the following exercises, solve for $x$

$\log (5x+1)=\log (x+3)+\log 2$

Short answer

By solving the equation $\log (5x+1)=\log (x+3)+\log 2$, the value of $x$is $\frac{5}{3}$

Step-by-step solution

Step 1. Given

An expression$\log (5x+1)=\log (x+3)+\log 2$

To solve the expression for$x$

Step 2. Use the Product property

By Product Property of logarithm,

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

$$\begin{gathered} \log (5x+1)=\log (x+3).\left(2\right) \\ \log (5x+1)=\log (2x+6) \end{gathered}$$

Step 3. Use One to One property

If ${\log }_{a}M={\log }_{a}N$, then $M=N$

$$\begin{gathered} 5x+1=2x+6 \\ 5x-2x=6-1 \\ 3x=5 \\ x=\frac{5}{3} \end{gathered}$$

The value of $x$ is $\frac{5}{3}$

Step 4. Check the solution

Substitute $x=\frac{5}{3}$in the original equation and solve.

$$\begin{gathered} \log \left(5\right(\frac{5}{3})+1)=\log \left(\right(\frac{5}{3})+3)+\log 2 \\ \log (\frac{25}{3}+1)=\log \left(\frac{14}{3}\right).\left(2\right) \\ \log \frac{28}{3}=\log \frac{28}{3} \end{gathered}$$

Hence the equation has been checked.