1. Let's state the problem: You want to understand why one of the $\ln x$ terms disappears when the numerator becomes $3x$ in an expression involving logarithms. 2. Recall the logarithm property: $\ln a + \ln b = \ln(ab)$ and $\ln a - \ln b = \ln\left(\frac{a}{b}\right)$. This means logarithms can be combined or separated by multiplication or division inside the log. 3. Suppose you start with an expression like $\frac{\ln(3x)}{\ln x}$. Using the property $\ln(3x) = \ln 3 + \ln x$, rewrite the numerator: $$\frac{\ln 3 + \ln x}{\ln x}$$ 4. Now split the fraction: $$\frac{\ln 3}{\ln x} + \frac{\ln x}{\ln x}$$ 5. Simplify the second term: $$\frac{\ln x}{\ln x} = 1$$ 6. So the expression becomes: $$\frac{\ln 3}{\ln x} + 1$$ 7. Notice that the $\ln x$ in the numerator has been separated and canceled in the second term, which is why it "disappears" or simplifies to 1. This is a direct consequence of the logarithm addition property and fraction simplification. Final answer: The $\ln x$ disappears because $\ln(3x)$ expands to $\ln 3 + \ln x$, and when divided by $\ln x$, the $\ln x$ terms simplify to 1.