Problem 85 - 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. $$\log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right)$$

Step-by-Step Solution

Answer

The given statement is true as it correctly applies the laws of logarithms.

Step 1: Analyze the Equation

Evaluate the logarithmic expression on the right hand side of the equation which is \( \log_{6}(x-1)-\log_{6}\left(x^{2}+4\right) \) to see whether it results in \( \log_{6} \left( \frac{x-1}{x^{2}+4} \right) \)

Step 2: Apply Logarithmic Laws

On applying the law of logarithms that the logarithm of the division of two numbers is equal to the subtraction of the logarithm of the two numbers, we get \( \log_{6}(x-1)-\log_{6}\left(x^{2}+4\right) = \log_{6} \left( \frac{x-1}{x^{2}+4} \right) \)

Step 3: Conclusion

After applying the laws, it can be concluded that the original statement is true and does not need any changes. This is because the right hand side of the equation and the left hand side of the equation are indeed equal when applying logarithmic laws.