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The Natural Logarithmic Function: Differentiation, Lecture notes of Calculus

NOTE: When using the properties of logarithms to rewrite logarithmic functions, check that the domain of the rewritten function is the same as the domain of ...

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2021/2022

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TheNaturalLogarithmicFunction:Differentiation
DEFINITIONOFTHENATURALLOGARITHMICFUNCTION
PropertiesoftheNaturalLogFunction
1.Thedomainisandtherangeis.
2.Thefunctioniscontinuous,increasing,andone‐to‐one.
3.Thegraphisconcavedownward.
If
a
and
b
arepositivenumbersand
n
isrational,then
thefollowingaretrue:
1.
2.
3.
4.
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pf4
pf5
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pf9
pfa

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The Natural Logarithmic Function: Differentiation

DEFINITION OF THE NATURAL LOGARITHMIC FUNCTION Properties of the Natural Log Function

  1. The domain is and the range is.
  2. The function is continuous, increasing, and one‐to‐one.
  3. The graph is concave downward. If a and b are positive numbers and n is rational, then the following are true:

NOTE: When using the properties of logarithms to rewrite logarithmic functions, check that the domain of the rewritten function is the same as the domain of the original. Consider and

Domain of is all reals except

while the domain of (^) is all positive real numbers.

EX #1: Sketch the graph of the function and state its domain.

EX #2: Find the slope of the tangent line to the graph of the logarithmic function at the point (1,0). Given:

EX #4: Find an equation of the tangent line to the graph of at the point (1, 2). 3 2 1 0 1 2 3 4 5 6 2 1 1 2 3 4 5 6 7 x y

EX #5: Find dy/dx by implicit differentiation.

EX #7: Find dy/dx using logarithmic differentiation