Problem 86 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{6}[4(x+1)]=\log _{6} 4+\log _{6}(x+1)$$

Step-by-Step Solution

Answer

The given equation is true as it abides by the logarithmic properties.

Step 1: Analyze the Given Equation

We are given \( \log _{6}[4(x+1)]=\log _{6} 4+\log _{6}(x+1)\). Let's examine if this equation holds true using the properties of logarithms.

Step 2: Apply the Logarithmic Properties

According to the logarithmic properties, \(\log_b mn = \log_b m + \log_b n\), where m and n are two numbers, and b is the base of the logarithm. If we apply this rule to the given equation, we find it aligns perfectly. So the mentioned statement is true.