# How do you use the tangent line approximation to approximate the value of ln(1004) ?

Oct 28, 2014

A formula for a tangent line approximation of a function f, also called linear approximation , is given by

$f \left(x\right) \approx f \left(a\right) + f ' \left(a\right) \left(x - a\right) ,$
which is a good approximation for $x$ when it is close enough to $a$.

I'm not sure, but I think the question is about approximate the value $\ln \left(1.004\right)$. Could you verify it please? Otherwise we will need to know an approximation to $\ln \left(10\right) .$

In this case, we have $f \left(x\right) = \ln \left(x\right)$, $x = 1.004$ and $a = 1$.

Since,
$f ' \left(x\right) = \frac{1}{x} \implies f ' \left(1\right) = 1$ and $\ln \left(1\right) = 0$, we get

$\ln \left(1.004\right) \approx \ln \left(1\right) + 1 \cdot \left(1.004 - 1\right) = 0.004 .$