Problem 83 - 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. $$\log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)}$$

Step-by-Step Solution

Answer

The given equation is false. The corrected equation is \( \log (x+3) - \log (2x) = \log \frac {(x+3)}{(2x)} \)

Step 1: Analyze the Equation

The given equation is \( \log (x+3)-\log (2x)=\frac{\log (x+3)}{\log (2x)} \). Our task is to determine if this equation is true or false.

Step 2: Apply logarithm subtraction rule on left side

We can apply the logarithm subtraction rule which states that \( \log_b a - \log_b c = \log_b (a/c) \). Applying this on the left side of the equation we get \( \log \frac{(x+3)}{(2x)} \).

Step 3: Compare both sides of the equation

We can now compare this to the result on the right side of the equation. It's clear now that \( \log \frac{(x+3)}{(2x)} \) is not equal to \( \frac{\log(x+3)}{ \log(2x) } \) thus indicating that the given equation is false.

Step 4: Correct the equation

To correct the equality, we must change either the left side or the right side. An obvious and simple change would be to write the left-side expression on the right side of the equals sign. This would result in \( \log (x+3) - \log (2x) = \log \frac {(x+3)}{(2x)} \).