Chapter 3, Problem 136RE

To determine

To determine whether the below equation is true or false.

  $\ln \left(x+y\right)=\ln \left(xy\right)$

Answer to Problem 136RE

The given statement is false.

Explanation of Solution

Given:

  $\ln \left(x+y\right)=\ln \left(xy\right)$

Concept Used:

Product property of logarithmic function: $\ln uv=\ln u+\ln v$

  $\ln u=x\iff u=e^x$

  $e^{\ln x}=x$

Calculation:

To determine whether the below equation is true or false.

  $\ln \left(x+y\right)=\ln \left(xy\right)$

First take the right side and simplify it using product property of logarithmic function, as

  $\ln \left(xy\right)=\ln x+\ln y$

Now, it is need not be true for every x and y that

  $\ln x+\ln y=\ln \left(x+y\right)$

Thus, left hand of the given equation is not equals to right hand.

Alternatively this can be proven using property $\ln u=x\iff u=e^x$ . As,

  $\begin{array}{c}\ln \left(x+y\right)=\ln \left(xy\right)\\ \Rightarrow x+y=e^{\ln \left(xy\right)}\\ \Rightarrow x+y=xy\text{ since }{e}^{\ln x}=x\end{array}$

But this is a contradiction.

Thus, the given statement is false.