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Ch. 7: Transcendental 7.2 Natural Logarithms
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Inverse functions:
= =
+
Solve for : =
= Switch and :
= =
Conseder = is an exponential function.
The inverse of exponential called a logarithm function
= = log
?
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Ch. 7: Transcendental 7.2 Natural Logarithms
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The most commonly used bases for logs are 10and e:
= and =
Natural logarithm
functionNatural common
function
A logarithm is the inverse of an
exponential function so that if
f(x) = ax, then f1(x) = logax.
The graph ofy= logax is
therefore a reflection ofy
= ax in the line y= x.
What is the e?
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Ch. 7: Transcendental 7.2 Natural Logarithms
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The number e is a famous
irrational number, and the first
few digits are:2.718281828459045235360287
4713527..
It is often called Euler's
number after Leonhard Euler.e is the base of the natural
logarithms (invented by John
Napier).
The natural logarithm function is not defined for negative values
of x.
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Ch. 7: Transcendental 7.2 Natural Logarithms
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Ch. 7: Transcendental 7.2 Natural Logarithms
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Example
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Ch. 7: Transcendental 7.2 Natural Logarithms
Example
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Ch. 7: Transcendental 7.2 Natural Logarithms
Example
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Ch. 7: Transcendental 7.2 Natural Logarithms
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Ch. 7: Transcendental 7.2 Natural Logarithms
Example
The derivatives of positive functions given by formulas that
involve products, quotients, and powers can often be found
more quickly if we take the natural logarithm of both sidesbefore differentiating.
Logarithmic Differentiation
The process, called logarithmic differentiation, is illustrated in
the next example.
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Ch. 7: Transcendental 7.2 Natural Logarithms
Example
Solution We take
natural logarithm of
both sides and
simplify the result
with the properties of
logarithms.
Then derivatives of
both sides with
respect tox:
Solve for dy/dx:
Substitute
for y
C. W 7.2:
2, 10, 26,
34, 44, 52,64, 66
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
Solution
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
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Ch. 7: Transcendental 7.2 Natural Logarithms
Solution
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For this section you asked to solve exercises1 to 68
Ch. 7: Transcendental 7.2 Natural Logarithms