Question

In Exercises, determine whether the statement is true or false given that $f(x)=\ln x.$ If it is false, explain why or give an example that shows it is false. $$f(x-2)=f(x)-f(2),x>2$$

Short answer

The given statement $f(x-2)=f(x)-f(2)$ is false for the function $f(x)=\ln x$. This is because when the values are substituted and simplified in the equation, it results in a false statement.

Step-by-step solution

Understand the Function

The function here is $f(x)=\ln x$. The logarithm function has a property that states $\ln a-\ln b=\ln \left(a/b\right)$. This critical property of logarithms will aid in solving this exercise.

Substitute the Values

Let's substitute the values of $x-2$ and $x$ in the function $f(x)=\ln x$. So, the statement $f(x-2)=f(x)-f(2)$ becomes $\ln (x-2)=\ln x-\ln 2$.

Simplify the Equation

Using the logarithm property $\ln a-\ln b=\ln \left(a/b\right)$, we can simplify $\ln x-\ln 2$ to $\ln \left(x/2\right)$. So, our equation becomes $\ln (x-2)=\ln \left(x/2\right)$.

Analyze the Equation

here comparing both sides, it's clear that $x-2$ does not equal to $x/2$ for values $x>2$. Thus, the given statement is false. An example would be $x=3$: substituting this into both sides results in $\ln 1$ on the left side and $\ln 1.5$ on the right side, which are not equal.