Questions: Determine whether the equation is true or false. Where possible, show work to produce a true statement. log (x+5)-log (6 x)= (log (x+5))/(log (6 x)) Select the correct choice below and, if necessary, fill in the answer box to complete A. The equation is false. The correct equation is log (x+5)-log (6 x)=log. B. The equation is true.

Solution Steps

To determine whether the given equation is true or false, we can use properties of logarithms. Specifically, we can use the property that \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\). We will apply this property to the left side of the equation and compare it to the right side. If both sides are equal for all valid \(x\), the equation is true; otherwise, it is false.

We start with the left side of the equation: \[ \log(x + 5) - \log(6x) \] Using the property of logarithms, we can combine the logs: \[ \log\left(\frac{x + 5}{6x}\right) \]

The right side of the equation is: \[ \frac{\log(x + 5)}{\log(6x)} \] This expression does not simplify directly to a logarithmic form that can be easily compared to the left side.

We have: \[ \log\left(\frac{x + 5}{6x}\right) \quad \text{(left side)} \] and \[ \frac{\log(x + 5)}{\log(6x)} \quad \text{(right side)} \] Since the left side simplifies to a single logarithmic expression while the right side remains a fraction of logarithms, we conclude that these two expressions are not equal for all valid \(x\).

Final Answer