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Chapter 3: Problem 97

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. $$\log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)}$$

Short Answer

Expert verified
The original statement is false. The true statement is: \(\log (x+3)-\log (2 x)=\log{\frac{(x+3)}{2x}}\).

Step by step solution

01

Apply the properties

Start by applying the rules of logarithms to the left side of the equation, providing: \(\log{\frac{(x+3)}{2x}}\)
02

Compare Both Sides

Compare both the simplified left side and the given right side \(\frac{\log (x+3)}{\log (2 x)}\). It is clear that these two expressions aren't the same, therefore the given equation is not true.
03

Correct the Equation

To fix the given equation, it would be correct if the given was \(\log (x+3)-\log (2 x)=\log{\frac{(x+3)}{2x}}\), that is, if the right side was simply the simplified form of the left side following the properties of logarithms.

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