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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!

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:

Product Rule

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