Using differentials, find the approximate values of the following:
loge10.02, it being given that loge10 = 2.3026
loge10.02, it being given that loge10 = 2.3026
Let us assume that f(x) = logex
Also, let x = 10 so that x + Δx = 10.02
⇒ 10 + Δx = 10.02
∴ Δx = 0.02
On differentiating f(x) with respect to x, we get
We know
When x = 10, we have
Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as
Here, and Δx = 0.02
⇒ Δf = (0.1)(0.02)
∴ Δf = 0.002
Now, we have f(10.02) = f(10) + Δf
⇒ f(10.02) = loge10 + 0.002
⇒ f(10.02) = 2.3026 + 0.002
∴ f(10.02) = 2.3046
Thus, loge10.02 ≈ 2.3046