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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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Unit 6

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!




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:




Example2
Example

Product Rule






Integrating is going backwards

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

Integrating is going backwards



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