Cambridge Additional Mathematics

y

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y=logz_x

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Logarithms (Chapter 5) 149

Consider the general exponential function f(x)=ax, a> 0 , a 6 =1.

The graph of y=ax is:

For a> 1 : For 0 <a< 1 :

Thehorizontal asymptotefor all of these functions is thex-axis y=0.

The inverse functionf¡^1 is given by x=ay,soy= logax.

If f(x)=ax where a> 0 , a 6 =1, then f¡^1 (x) = logax.

Since f¡^1 (x) = logax is an inverse function, it is a reflection of f(x)=ax in the line y=x.We

may therefore deduce the following properties:

Function f(x)=ax f¡^1 (x) = logax

Domain fx:x 2 Rg fx:x> 0 g

Range fy:y> 0 g fy:y 2 Rg

Asymptote horizontal y=0 vertical x=0

The graph of y= logax for a> 1 : The graph of y= logax for 0 <a< 1 :

Thevertical asymptoteof y= logax is they-axis x=0.

Since we can only find logarithms of positive numbers, the domain of f¡^1 (x) = logax is fxjx> 0 g.

In general, y= loga(g(x)) is defined when g(x)> 0.

H Graphs of logarithmic functions

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4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_05\149CamAdd_05.cdr Monday, 23 December 2013 1:40:45 PM BRIAN