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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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
(x 0, e 2.718281)
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.
Finding the anti-derivative using natural logs is fun, fun, fun