Question

In the following exercises,solve each logarithmic equation.

${\log }_3(5x-4)=4$

Short answer

The solution is$x=17$.

Step-by-step solution

Step 1.Given information

Consider the logarithmic equation ${\log }_3(5x-4)=4$.

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

Step 2.Solve the given logarithmic equation

Comparing ${\log }_3(5x-4)=4$with the standard form $y={\log }_{a}x$,you have $y=4,a=3,x=5x-4$

Convert logarithmic equation in exponential equation as shown below

$$\begin{gathered} x=a^y \\ 5x-4=3^4 \\ 5x-4=81 \end{gathered}$$

So,the solution is $5x-4=81$

Now solve the equation as

$$\begin{gathered} 5x-4+4=81+4[add4] \\ 5x=85[divideby5] \\ x=17 \end{gathered}$$

So,the solution is$x=17$

Step 3.Check

To check the solution,replace a by 17 in original equation

$$\begin{gathered} {\log }_3(5.17-4)=4[x=17] \\ {\log }_{3}81=4 \\ {3}^4=81 \\ 81=81\left[true\right] \end{gathered}$$

Therefore,the solution is$x=17$