Solution Found!
The natural logarithm function, ln, is defined so that eln
Chapter 2, Problem 12P(choose chapter or problem)
The natural logarithm function, ln, is defined so that \(e^{\ln x}=x\) for any positive number x.
(a) Sketch a graph of the natural logarithm function.
(b) Prove the identities
\(\ln a b=\ln a+\ln b \quad \text { and } \quad \ln a^{b}=b \ln a\)
(c) Prove that \(\frac{d}{d x} \ln x=\frac{1}{x}\)
(d) Derive the useful approximation
\(\ln (1+x) \approx x\),
which is valid when \(|x| \ll 1\). Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.
Questions & Answers
QUESTION:
The natural logarithm function, ln, is defined so that \(e^{\ln x}=x\) for any positive number x.
(a) Sketch a graph of the natural logarithm function.
(b) Prove the identities
\(\ln a b=\ln a+\ln b \quad \text { and } \quad \ln a^{b}=b \ln a\)
(c) Prove that \(\frac{d}{d x} \ln x=\frac{1}{x}\)
(d) Derive the useful approximation
\(\ln (1+x) \approx x\),
which is valid when \(|x| \ll 1\). Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.
ANSWER:ANSWER:
a)
The natural logarithm function is,
.
The graph is shown below.
b)
i)
.
Proof:
We know that,
------------------(1)
We are going to prove it for any arbitrary base which can be changed to e later.
Step 1:-
let’s take.
.
By the definition of logarithm.