Page 279 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes logarithms. Though the following properties and methods are true for a logarithm of any base,
only the natural logarithm (base e, where e 2.718281828), ln, will be used in this problem set.

10.2.1 Properties of the Natural Logarithm

1. ln 1 = 0

2. ln e = 1
x
3. ln e = x
x
4. ln y = x ln y
5. ln (xy) = ln x + ln y

x
6. ln = ln x ln y
y

10.2.2 Avoid the Following List of Common Mistakes

1. ln (x + y) = ln x + ln y
2. ln (x y) = ln x ln y
3. ln (xy) = ln x ln y

x ln x
4. ln
y ln y
lnx
5. lnx ln y
ln y
The following exaples range in difficulty from average to challenging:

Example: Differentiate y = x x
Solution:
Because a variable is raised to a variable power in this function, the ordinary rules of
differentiation do not apply ! The function must first be revised before a derivative can be taken.
Begin with
y = x x
Apply the natural logarithm to both sides of this equation getting
ln y = ln x x
= x ln x

Differentiate both sides of this equation. The left-hand side requires the chain rule since y
represents a function of x. Use the product rule on the right-hand side. Thus, beginning with
ln y = x ln x

and differentiating, we get
1 1
y = x (1)lnx
y x
= 1 + ln x

272 LOVELY PROFESSIONAL UNIVERSITY