Approximating $\ln(1+2x) = 2x$ using linear approximation

Question

The question is as follows:

Use the linear approximation formula
$$\boxed{f(x+\Delta x) \approx f(x) +f'(x)\Delta x}$$ to show that $\ln(2x+1) \approx 2x$ for small values of $x$.

I am having trouble understanding what $\Delta x$ is supposed to be in the formula given.

Also should $f(x) = \ln(1+2x)$ or $f(x) = \ln(1+x)$. Will that make any difference?

Answer 1

Let's rewrite the boxed identity using a different variable:

$$\boxed{f(z + \Delta z) \approx f(z) + f'(z) \Delta z}.$$

Now choose $$f(x) = \log x = \ln x, \\ z = 1, \\ \Delta z = 2x.$$