Unit 6
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Unit 6. Natural Logs. A logarithm is an exponent!. For x  0 and 0  a  1, y = log a x if and only if x = a y . The function given by f ( x ) = log a x i s called the logarithmic function with base a . Every logarithmic equation has an equivalent exponential form:

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Natural Logs

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Unit 6

Unit 6

Natural Logs


Natural logs

A logarithm is an exponent!

For x 0 and 0  a  1,

y = loga x if and only if x = ay.

The function given by f(x) = loga x is called the logarithmic function with base a.

Every logarithmic equation has an equivalent exponential form:

y= loga x is equivalent to x =ay

A logarithmic function is the inverse function of an exponential function.

Exponential function:y = ax

Logarithmic function:y = logax is equivalent to x = ay


Natural logs

y

(x  0, e 2.718281)

x

5

–5

y = ln x is equivalent to ey = x

y = ln x

The function defined by f(x) = logex = ln x

is called the natural logarithm function.

In Calculus, we work almost exclusively with natural logarithms!


Examples

Examples


Examples1

Examples


Natural logs

Derivative of Logarithmic Functions

The derivative is

Notice that the derivative of expressions such as ln|f(x)| has no logarithm in the answer.

Example:

Solution:


Example

Example


Example1

Example


Example2

Example

Product Rule


Example3

Example


Example4

Example


Example5

Example


Example6

Example


Integrating is going backwards

Finding the anti-derivative using natural logs is fun, fun, fun 

Integrating is going backwards


Integrals of 6 basic trig functions

Integrals of 6 basic trig functions


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