Unit 6 Lesson 1 Natural Logs

y = loga x if and only if x = a y. For x  0 and 0  a  1, y = loga x if and only if x = a y. 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 = a y A logarithm is an exponent! A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay

In Calculus, we work almost exclusively with natural logarithms! 5 –5 y = ln x The function defined by f(x) = loge x = ln x (x  0, e 2.718281) is called the natural logarithm function. y = ln x is equivalent to e y = x In Calculus, we work almost exclusively with natural logarithms!

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Examples

Example: Solution: 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:

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Example Product Rule

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

Integrals of 6 basic trig functions