Chapter 6, Problem 11CT

To determine

To write : the similarities and differences between the equations $4^{5x-2}=16$ and ${\log }_4\left(10x+6\right)=1$ . Then solve each equations

Answer to Problem 11CT

The 1^st and 2^nd equation solution is $x=\frac{4}{5}$ and $x=\frac{-1}{5}$ .

Explanation of Solution

Given information The equations $4^{5x-2}=16$ and ${\log }_4\left(10x+6\right)=1$

The similarities in solving the equations, the base is 4 and the differences between them is that the equation $4^{5x-2}=16$ is an exponential equation and ${\log }_4\left(10x+6\right)=1$ is a logarithmic equation.

1^st equation

  $\begin{array}{l}{4}^{5x-2}=16\\ 4^{5x-2}={\left(4\right)}^2\\ 5x-2=2\\ 5x-2+2=24+2\\ 5x=4\\ x=\frac{4}{5}\end{array}$

2^nd equation

  $\begin{array}{l}{\log }_4\left(10x+6\right)=1\\ \left(10x+6\right)=4^1\\ 10x+6=4\\ 10x=4-6\\ 10x=-2\\ x=\frac{-2}{10}\\ x=\frac{-1}{5}\end{array}$

Hence, the 1^st and 2^nd equation solution is $x=\frac{4}{5}$ and $x=\frac{-1}{5}$ .