5. Recall that we defined In x = " dt. for x > 0. Prove that In(xy) = In(x) + In(y) for all x,y > 0. (*) Hint: Fix y > 0 and view both sides of (*) as functions of x. Now argue using the result of Problem 4.
5. Recall that we defined In x = " dt. for x > 0. Prove that In(xy) = In(x) + In(y) for all x,y > 0. (*) Hint: Fix y > 0 and view both sides of (*) as functions of x. Now argue using the result of Problem 4.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
(use the result of this, do not need to prove) *Problem 4: Suppose f and g are differentiable functions on an interval I. Show that if f′(x) = g′(x) for all x ∈ I and f(x0) = g(x0) for some x0 ∈ I, then f(x) = g(x) for all x ∈ I.
![**Problem 5:**
Recall that we defined \( \ln x = \int_1^x \frac{dt}{t} \) for \( x > 0 \). Prove that
\[
\ln(xy) = \ln(x) + \ln(y) \quad \text{for all } \; x, y > 0. \quad (\ast)
\]
*Hint:* Fix \( y > 0 \) and view both sides of \( (\ast) \) as functions of \( x \). Now argue using the result of Problem 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32b90586-0d73-43dc-903b-9a80d199c8ec%2Fd2f88dc4-9e52-45e4-991e-a138487774c9%2Frq0568a_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5:**
Recall that we defined \( \ln x = \int_1^x \frac{dt}{t} \) for \( x > 0 \). Prove that
\[
\ln(xy) = \ln(x) + \ln(y) \quad \text{for all } \; x, y > 0. \quad (\ast)
\]
*Hint:* Fix \( y > 0 \) and view both sides of \( (\ast) \) as functions of \( x \). Now argue using the result of Problem 4.
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