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Fun with logarithms.(a) Simplify the expression ea ln b.
Chapter 2, Problem 13P(choose chapter or problem)
Problem 13P
Fun with logarithms.
(a) Simplify the expression ea ln b. (That is, write it in a way that doesn’t involve logarithms.)
(b) Assuming that b ≪ a. prove that ln(a + b) ≈ (ln a) + (b/a). (Hint: Factor out the a from the argument of the logarithm, so that you can apply the approximation of part (d) of the previous problem.)
Problem:
The natural logarithm function, ln, is defined so that eln x = x for any positive number x:
(a) Sketch a graph of the natural logarithm function.
(b) Prove the identities
ln ab = ln a + ln b and ln ab = b ln a.
(c) Prove that
(d) Derive the useful approximation
ln(1 + x) ≈ x,
which is valid when |x| ≪ 1. Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.
Questions & Answers
QUESTION:
Problem 13P
Fun with logarithms.
(a) Simplify the expression ea ln b. (That is, write it in a way that doesn’t involve logarithms.)
(b) Assuming that b ≪ a. prove that ln(a + b) ≈ (ln a) + (b/a). (Hint: Factor out the a from the argument of the logarithm, so that you can apply the approximation of part (d) of the previous problem.)
Problem:
The natural logarithm function, ln, is defined so that eln x = x for any positive number x:
(a) Sketch a graph of the natural logarithm function.
(b) Prove the identities
ln ab = ln a + ln b and ln ab = b ln a.
(c) Prove that
(d) Derive the useful approximation
ln(1 + x) ≈ x,
which is valid when |x| ≪ 1. Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.
ANSWER:
ANSWER:
- The expression can be simplified in this manner,
b) The assumption is, b << a.
Step 1:-
So,
-----------------------(1)
Now by expanding the 2nd term in the right hand side,
The higher order terms get smaller and smaller as b is extremely small than a.
So, up to the 1st order,
--------------------(2)