lecture05 - Lecture 5 - Logarithms, Slope of a Function,...

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Lecture 5 - Logarithms, Slope of a Function, Derivatives 5.1 Logarithms Note the graph of e x This graph passes the horizontal line test, so f ( x ) = e x is one-to-one and therefore has an inverse function. This is also true of f ( x ) = a x for any a > 0 ,a 6 = 1. More generally, for any a > 1 the graph of a x and its inverse look like this. If f ( x ) = a x , then we define the inverse function f - 1 to be the logarithm with base a , and write f - 1 ( x ) = log a ( x )
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Note that, since the image of a x is only the positive numbers, the domain of log a ( x ) is all positive real numbers. The key property is: log a x = b ⇐⇒ a b = x Examples: log 10 10 = 1 10 ? = 10 log 5 25 = 2 5 ? = 25 log 4 1 2 = - 1 2 4 ? = 1 2 log 5 1 125 = - 3 5 ? = 1 125 log equation corresponding exponential equation Log Rules 1. Most important: by the properties of inverse functions we have log b ( b x ) = x and b log b x = x The most important case of logs is when b = e . Log base e has a special name, in fact we define log e x = ln( x ). So the above becomes ln( e x ) = x and e ln( x ) = x LEARN THIS!! The function ln( x ) is known as the natural logarithm function , and ln( x ) should be read as “the natural logarithm of x ”. In class, you may also hear me read this as “lawn x ”, but this isn’t as standard. Other rules: (I will state for ln, but they work for every log). Suppose that x,y > 0 2. ln( xy ) = ln( x ) + ln( y ) 3. ln ± x y ² = ln x - ln y 4. ln( x y ) = y ln( x ) Calculations: e 3ln( x ) = e ln( x 3 ) = x 3 ln ³ 1 e ´ = ln( e - 1 ) = - 1 Rewrite the following: ln ± xy z ² = ln( xy ) - ln( z ) = ln( x ) + ln( y ) - ln( z )
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lecture05 - Lecture 5 - Logarithms, Slope of a Function,...

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