Estimate ln(1.05) with a linear approximation

Step by step

The question

Question: estimate ln 1.05 with a linear approximation.

The function is f(x) = ln x.

Near 1.05 there is a friendly number: a = 1.

ln 1 is 0.

1 The point

Put 1 into f.

We get 0.

The point is (1, 0).

2 The slope there

The slope is the derivative: f′(x) = 1 over x.

At x = 1, the slope is 1.

3 The tangent line

The tangent line: start at the height f of a, and go up by the slope times the distance from a.

So L(x) = x − 1.

Let's see it.

The curve.

And the tangent line at x = 1.

Near 1, they are almost the same.

4 Estimate

Put 1.05 into L.

Careful!

Use the distance from 1: 1.05 minus 1.

Not 1.05 itself.

1.05 minus 1 is 0.05.

So ln 1.05 ≈ 0.05.

5 Check

Let's check with a calculator: ln 1.05 is 0.04879 and a bit.

Our estimate 0.05 is very close.

Right!

Let's look back.

First: the point.

Then: the slope there.

Then: the tangent line.

Then: estimate.

Last: check.