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Unit 6

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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 is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function is the inverse function of an exponential function. Exponential function:y = a x Logarithmic function:y = log a x is equivalent to x = a y A logarithm is an exponent!

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

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Derivative of Logarithmic Functions The derivative is Example: Solution: Notice that the derivative of expressions such as ln|f(x)| has no logarithm in the answer.

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Product Rule

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

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Solving Exponential Equations. We can solve exponential equations using logarithms. By converting to a logarithm, we can move the variable from the exponent.

Solving Exponential Equations. We can solve exponential equations using logarithms. By converting to a logarithm, we can move the variable from the exponent.

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