Question

In the following exercises,solve each logarithmic equation.

${\log }_2\left(5x+1\right)=4$

Short answer

The solution is$x=3$.

Step-by-step solution

Step 1.Given information

Consider the logarithmic equation ${\log }_2\left(5x+1\right)=4$

The logarithmic from $y={\log }_{a}x$is equivalent to exponential form$x=a^y$.

Step 2.Solve the given logarithmic equation.

Comparing ${\log }_2(5x+1)=4$with the standard form $y={\log }_{a}x$,you have$y=4,a=2,x=5x+1$

Convert logarithmic equation in exponential equation as shown below

$$\begin{gathered} x=a^y \\ 5x+1=2^4 \\ 5x+1=16 \end{gathered}$$

So,the solution is $5x+1=16$

Now solve the equation as

$$\begin{gathered} 5x+1-1=16-1[substract1] \\ 5x=15 \\ \frac{5x}{5}=\frac{15}{5}[divideby5] \\ x=3 \end{gathered}$$

So,the solution is$x=3$

Step 3.Check

To check the solution,replace a by 3 in original equation

$$\begin{gathered} {\log }_2(5.3+1)=4[x=3] \\ {\log }_{2}16=4 \\ {2}^4=16 \\ 16=16\left[true\right] \end{gathered}$$

Therefore,the solution is$x=3$