Chapter 3.4, Problem 87E

To determine

Write the given verbal statement in an equation form and check whether the statement is true or false.

Answer to Problem 87E

The given statement is true.

Explanation of Solution

Given:

The given statement is that the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Calculation:

The equation form of the given statement is

  ${\log }_x\left(ab\right)={\log }_{x}a+{\log }_{x}b$

  $a$ and $b$ is the two numbers.

Proof of the statement is

Let

  $p={\log }_{x}a\Rightarrow a=x^p$

And

  $q={\log }_{x}b\Rightarrow b=x^q$

From LHS

  $\begin{array}{l}{\log }_x\left(ab\right)\\ ={\log }_x\left(x^p\cdot x^q\right)\\ ={\log }_x\left(x^{p+q}\right)\\ =\left(p+q\right){\log }_{x}x\\ =p+q\left[{\log }_{x}x=1\right]\end{array}$

From RHS

  $\begin{array}{l}{\log }_{x}a+{\log }_{x}b\\ ={\log }_x{x}^p+{\log }_x{x}^q\\ =p{\log }_{x}x+q{\log }_{x}x\\ =p+q\left[{\log }_{x}x=1\right]\end{array}$

  $\text{LHS}=\text{RHS}$

Hence the given statement is true.