7.2 Natiral Logarithms

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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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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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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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Example


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Example


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Example


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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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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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Solution


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Solution

Solution


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Solution


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Solution


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Solution


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Solution


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Solution


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For this section you asked to solve exercises1 to 68


7.2 Natiral Logarithms

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