Chapter 3.4, Problem 5E

(a)

To determine

To find: Whether the given value of x is a solution of the equation or not.

Answer to Problem 5E

The x -value, $x=1021$ is solution of the equation ${\log }_2\left(x+3\right)=10$ .

Explanation of Solution

Given information:

The equation ${\log }_2\left(x+3\right)=10$ and x -value $x=1021$ .

Formula used:

Substitute the provided value of the variable in the function to check whether left hand side of the equation is equal to right hand side of the equation.

The inverse property of logarithmic function that is ${\log }_a{a}^x=x$ .

Calculation:

Consider the provided equation ${\log }_2\left(x+3\right)=10$ and x -value that is $x=1021$ .

Substitute x as 1021 in the left hand side of the above equation,

  $\begin{array}{c}{\log }_2\left(x+3\right)=10\\ {\log }_2\left(1021+3\right)=10\\ {\log }_{2}1024=10\\ {\log }_2{2}^{10}=10\end{array}$

Recall the inverse property of logarithmic function that is ${\log }_a{a}^x=x$ .

Apply it,

Simplify it further as,

  $10=10$

The above equation is true.

Therefore, $x=1021$ satisfy the equation ${\log }_2\left(x+3\right)=10$ .

Hence, the x -value, $x=1021$ is solution of the equation ${\log }_2\left(x+3\right)=10$ .

(b)

To determine

To find: Whether the given value of x is a solution of the equation or not.

Answer to Problem 5E

The x -value, $x=17$ is not a solution of the equation ${\log }_2\left(x+3\right)=10$ .

Explanation of Solution

Given information:

The equation ${\log }_2\left(x+3\right)=10$ and x -value $x=17$ .

Formula used:

Substitute the provided value of the variable in the function to check whether left hand side of the equation is equal to right hand side of the equation.

Base change formula for logarithmic function ${\log }_{a}x=\frac{\log x}{\log a}$ .

Calculation:

Consider the provided equation ${\log }_2\left(x+3\right)=10$ and x -value that is $x=17$ .

Substitute x as 17 in the left hand side of the above equation,

  $\begin{array}{l}{\log }_2\left(x+3\right)=10\\ {\log }_2\left(17+3\right)=10\\ {\log }_{2}21=10\end{array}$

Recall the formula of change base ${\log }_{a}x=\frac{\log x}{\log a}$ .

Apply it,

Simplify it further as,

  $\begin{array}{c}\frac{\log 21}{\log 2}=10\\ \frac{3.0445}{0.6931}=10\\ 4.392=10\end{array}$

The above equation is false as $4.392\ne 10$ .

Therefore, $x=17$ does not satisfy the equation ${\log }_2\left(x+3\right)=10$ .

Hence, the x -value, $x=17$ is not a solution of the equation ${\log }_2\left(x+3\right)=10$ .

(c)

To determine

To find: Whether the given value of x is a solution of the equation or not.

Answer to Problem 5E

The x -value, $x=10-3$ is a solution of the equation ${\log }_2\left(x+3\right)=10$ .

Explanation of Solution

Given information:

The equation ${\log }_2\left(x+3\right)=10$ and x -value $x={10}^2-3$ .

Formula used:

Substitute the provided value of the variable in the function to check whether left hand side of the equation is equal to right hand side of the equation.

Calculation:

Consider the provided equation ${\log }_2\left(x+3\right)=10$ and x -value that is $x={10}^2-3$ .

Substitute x as ${10}^2-3$ in the left hand side of the above equation,

  $\begin{array}{c}{\log }_2\left(x+3\right)=10\\ {\log }_2\left({10}^2-3+3\right)=10\\ {\log }_2{10}^2=10\\ 2{\log }_{2}10=10\end{array}$

Recall the formula of base change of logarithmic functions ${\log }_{a}x=\frac{\log x}{\log a}$ .

Apply it,

  $\begin{array}{c}2\cdot \frac{\log 10}{\log 2}=10\\ \frac{2.302}{0.693}=5\\ 3.32=5\end{array}$

The above equation is false as $3.32\ne 5$

Therefore, $x={10}^2-3$ does not satisfy the equation ${\log }_2\left(x+3\right)=10$ .

Hence, the x -value, $x={10}^2-3$ is a solution of the equation ${\log }_2\left(x+3\right)=10$ .