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

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


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


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


Example

Product Rule


Example


Example


Example


Example


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

Integrating is going backwards


Integrals of 6 basic trig functions


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