Chapter 3.4, Problem 57E

Solution

To determine: A given statement as the true or false.

The given statement is true.

Given data: The given statement is: the logarithm of the product of two positive numbers is the sum of the logarithms of the number.

Method/Formula used:

The logarithm of a number is defined as: If $c^m=p\left(\text{index}\text{form}\right)$ then its logarithms form is ${\log }_{c}p=m$

Explanation:

Let a and b are two numbers. Then to show that $\log \left(a\cdot b\right)=\log a\text{}+\log b$ .

Suppose $a=c^x$ and $b=c^y$ , then by the definition of log,

  $\begin{array}{l}{\log }_{c}a=x\\ {\log }_{c}b=y\end{array}$

Adding

  ${\log }_{c}a+{\log }_{c}b=x+y$ ... (1)

Now

  $\begin{array}{c}a\cdot b=c^x\cdot c^y\\ =c^{x+y}\\ c^{x+y}=a\cdot b\end{array}$

By the definition of log

  ${\log }_{c}a\cdot b=x+y$ … (2)

From results (1) and (2),

  ${\log }_{c}a+{\log }_{c}b={\log }_c\left(a\cdot b\right)$

Therefore, the statement the logarithm of the product of two positive numbers is the sum of the logarithms of the number, is true.