Chapter 4.5, Problem 2E

a.

To determine

To define: The equation of logarithms ,combine the logarithms on the LHS to get the equivalent equation.

Answer to Problem 2E

The logarithm equation $\log 3+\log \left(x+2\right)=\log x$ , and combine the logarithm to LHS to get the equivalent equation $\log \left(3\left(x-2\right)\right)=\log x$ .

Explanation of Solution

Given information:

The term standard-form logarithm.

The LHS of equation when combine the log’s on LHS from which get the resultant equation $\log \left(3\left(x-2\right)\right)=\log x$ .

It is the LHS equivalent standard-form .

The equation is, $\log 3+\log \left(x+2\right)=\log x$

Split to take common:

  $\begin{array}{c}\log 3+\log \left(x+2\right)=\log x\\ \log \left(xy\right)=\log x+\log y\\ \log \left(3\left(x-2\right)\right)=\log x\end{array}$

For example: The equation,

  $\log 2+\log \left(x+3\right)=\log x$

Apply the log identity, $\begin{array}{c}\log \left(xy\right)=\log x+\log y\\ \log 2+\log \left(x+3\right)=\log x\\ \log \left(2\left(x+3\right)\right)=\log x\end{array}$

b.

To determine

To define: The logarithmic equation, $\log 3+\log \left(x+2\right)=\log x$ is use to the fact that log is one_to_one to get the equation.

Answer to Problem 2E

The log is one_to_one to get the equivalent relation , $\begin{array}{c}3\left(x-2\right)=x\\ 3x-6=x\end{array}$ .

Explanation of Solution

Given information:

The logarithmic equation.

The equation is use to the fact that log is one_to_one to get the equivalence equation it means here check the equation is one_to_one and satisfied the logarithmic equation .

The equation is, $\log 3+\log \left(x+2\right)=\log x$

Split to take common:

  $\begin{array}{c}\log 3+\log \left(x+2\right)=\log x\\ \log \left(xy\right)=\log x+\log y\\ \log \left(3\left(x-2\right)\right)=\log x\\ 3x-6=x\end{array}$

For example: The equation,

  $\log 2+\log \left(x+3\right)=\log x$

Apply the log identity,

  $\begin{array}{c}\log \left(xy\right)=\log x+\log y\\ \log 2+\log \left(x+3\right)=\log x\\ \log \left(2\left(x+3\right)\right)=\log x\\ 2\left(x+3\right)=x\end{array}$

To find the value of x,

  $\begin{array}{c}2\left(x+3\right)=x\\ 2x-6=x\\ 2x-x=6\end{array}$