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Exponential and logarithm functions - Mathematics resources

Exponential andlogarithm functionsmc-TY-explogfns-2009-1 Exponential functions and logarithm functions are important in both theory and practice. In thisunit we look at the graphs of Exponential and logarithm functions , and see how they are order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: specify for which values ofathe Exponential functionf(x) =axmay be defined, recognize the domain and range of an Exponential function, identify a particular point which is on the graph of every Exponential function, specify for which values ofathe logarithm functionf(x) = logaxmay be defined, recognize the domain and range of a logarithm function, identify a particular point which is on the graph of every logarithm function, understand the relationship between the Exponential functionf(x) =exand the naturallogarithm functionf(x) = relationship between Exponential functions andlogarithm mathcentre 20091.

The important properties of the graphs of these types of functions are: •f(0) = 1 for all values of a. This is because a0 = 1 for any value of a. •f(x) > 0 for all values of a.

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Transcription of Exponential and logarithm functions - Mathematics resources

1 Exponential andlogarithm functionsmc-TY-explogfns-2009-1 Exponential functions and logarithm functions are important in both theory and practice. In thisunit we look at the graphs of Exponential and logarithm functions , and see how they are order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: specify for which values ofathe Exponential functionf(x) =axmay be defined, recognize the domain and range of an Exponential function, identify a particular point which is on the graph of every Exponential function, specify for which values ofathe logarithm functionf(x) = logaxmay be defined, recognize the domain and range of a logarithm function, identify a particular point which is on the graph of every logarithm function, understand the relationship between the Exponential functionf(x) =exand the naturallogarithm functionf(x) = relationship between Exponential functions andlogarithm mathcentre 20091.

2 Exponential functionsConsider a function of the formf(x) =ax, wherea >0. Such a function is called anexponentialfunction. We can take three different cases, wherea= 1,0< a <1anda > 1thenf(x) = 1x= this just gives us the constant functionf(x) = happens ifa >1? To examine this case, take a numerical example. Suppose thata= (x) = 2xf(0) = 20= 1f(1) = 21= 2f( 1) = 2 1= 1/21=12f(2) = 22= 4f( 2) = 2 2= 1/22=14f(3) = 23= 8f( 3) = 2 3= 1/23=18We can put these results into a table, and plot a graph of the (x) 318 214 11201122438f(x)xf(x) = 2xThis example demonstrates the general shape for graphs of functions of the formf(x) =axwhena > is the effect of varyinga? We can see this by looking at sketches of a few graphs of (x)xf(x) = 2xf(x) = 5xf(x) = mathcentre 2009 The important properties of the graphs of these types of functions are: f(0) = 1for all values ofa.

3 This is becausea0= 1for any value ofa. f(x)>0for all values ofa. This is becausea >0impliesax> happens if0< a <1? To examine this case, take another numerical example. Supposethata= (x) =(12)xf(0) =(12)0= 1f(1) =(12)1=(12)f( 1) =(12) 1=(21)1= 2f(2) =(12)2=(14)f( 2) =(12) 2=(21)2= 4f(3) =(12)3=(18)f( 3) =(12) 3=(21)3= 8We can put these results into a table, and plot a graph of the (x) 38 24 1201112214318f(x)xf(x) = ( )x12 This example demonstrates the general shape for graphs of functions of the formf(x) =axwhen0< a < is the effect of varyinga? Again we can see by looking at sketches of a few graphs ofsimilar (x)xf(x) = ( )x12f(x) = ( )x110f(x) = ( ) mathcentre 2009 The important properties of the graphs of these types of functions are: f(0) = 1for all values ofa.

4 This is becausea0= 1for any value ofa. f(x)>0for all values ofa. This is becausea >0impliesax> that these properties are the same as whena >1. One interesting thing that you mighthave spotted is thatf(x) = (12)x= 2 xis a reflection off(x) = 2xin thef(x)axis, and thatf(x) = (15)x= 5 xis a reflection off(x) = 5xin thef(x) (x)xf(x) = ( )x12f(x) = ( )x15f(x) = 2xf(x) = 5xIn general,f(x) = (1/a)x=a xis a reflection off(x) =axin thef(x) particularly important example of an Exponential function arises whena=e. You might recallthat the number e is approximately equal to The functionf(x) =exis often called the Exponential function. Since e>1and1/e<1, we can sketch the graphs of the exponentialfunctionsf(x) =exandf(x) =e x= (1/e) (x)xf(x) = exf(x) = e mathcentre 2009 Key PointA function of the formf(x) =ax(wherea >0) is called an Exponential functionf(x) = 1xis just the constant functionf(x) = functionf(x) =axfora >1has a graph which is close to thex-axis for negativexandincreases rapidly for functionf(x) =axfor0< a <1has a graph which is close to thex-axis for positivexand increases rapidly for decreasing any value ofa, the graph always passes through the point(0,1).

5 The graph off(x) =(1/a)x=a xis a reflection, in the vertical axis, of the graph off(x) = particularly important exponental function isf(x) =ex, where e= .. This is oftencalled the Exponential logarithm functionsWe shall now look at logarithm functions . These are functions of the formf(x) = logaxwherea >0. We do not consider the casea= 1, as this will not give us a valid happens ifa >1? To examine this case, take a numerical example. Suppose thata= (x) = log2xmeans2f(x)=x .An important point to note here is that, regardless of the argument,2f(x)>0. So we shallconsider only positive (1) = log21means2f(1)= 1sof(1) = 0f(2) = log22means2f(2)= 2sof(2) = 1f(4) = log24means2f(4)= 4sof(4) = 2f(12) = log2(12)means2f(12)=12= 2 1sof(12) = 1f(14) = log2(14)means2f(14)=14= 2 2sof(14) = 2We can put these results into a table, and plot a graph of the mathcentre 2009xf(x)14 212 1102142f(x)xf(x) = log2 xThis example demonstrates the general shape for graphs of functions of the formf(x) = logaxwhena > is the effect of varyinga?

6 We can see by looking at sketches of a few graphs of similarfunctions. For the special case wherea=e, we often writelnxinstead (x)xf(x) = log2 xf(x) = log5 xf(x) = loge x = ln xThe important properties of the graphs of these types of functions are: f(1) = 0for all values ofa; we must havex >0for all values happens if0< a <1? To examine this case, take another numerical example. Supposethata=12. Thenf(x) = log1/2xmeans(12)f(x)=x .An important point to note here is that, regardless of the argument,(12)f(x)>0. So we shallconsider only positive mathcentre 2009f(x) =(12)xf(1) = log1/21means(12)f(1)= 1sof(1) = 0f(2) = log1/22means(12)f(2)= 2 =(12) 1sof(2) = 1f(4) = log1/24means(12)f(4)= 4 =(12) 2sof(4) = 2f(12) = log1/2(12)means(12)f(12)=12sof(12) = 1f(14) = log1/2(14)means(12)f(14)=14=(12)2sof(14) = 2We can put these results into a table, and plot a graph of the (x)142121102 14 2f(x)xf(x) = log1/2 xThis example demonstrates the general shape for graphs of functions of the formf(x) = logaxwhen0< a < is the effect of varyinga?

7 Again we can see by looking at sketches of a few graphs ofsimilar (x)xf(x) = log1/2 xf(x) = log1/5 xf(x) = log1/e mathcentre 2009 The important properties of the graphs of these types of functions are: f(1) = 0for all values ofa; we must havex >0for all values interesting thing that you might well have spotted is thatf(x) = log1/5xis a reflection off(x) = log5xin thex-axisandf(x) = log1/2xis a reflection off(x) = log2xin (x)xf(x) = log1/2 xf(x) = log1/5 xf(x) = log2 xf(x) = log5 xGenerally,f(x) = log1/axis a reflection off(x) = logaxin PointA function of the formf(x) = logax(wherea >0anda6= 1) is called a logarithm functionf(x) = logaxfora >1has a graph which is close to the negativef(x)-axis forx <1and increases slowly for functionf(x) = logaxfor0< a <1has a graph which is close to the positivef(x)-axisforx <1and decreases slowly for any value ofa, the graph always passes through the point(1,0).

8 The graph off(x) = log1/axis a reflection, in the horizontal axis, of the graph off(x) = particularly important logarithm function isf(x) = logex, where e= .. This is oftencalled the natural logarithm function, and writtenf(x) = mathcentre 20093. The relationship between Exponential functions and log-arithm functionsWe can see the relationship between the Exponential functionf(x) =exand the logarithmfunctionf(x) = lnxby looking at their (x)xf(x) = ln xf(x) = exf(x) = xYou can see straight away that the logarithm function is a reflection of the Exponential functionin the line represented byf(x) =x. In other words, the axes have been swapped:xbecomesf(x), andf(x) PointThe Exponential functionf(x) =exis the inverse of the logarithm functionf(x) = Sketch the graph of the functionf(x) =axfor the following values ofa, on the same axes.

9 (a)a= 3(b)a= 6(c)a= 1(d)a=13(e)a=162. Sketch the graph of the functionf(x) = logaxfor the following values ofa, on the sameaxes.(a)a= 3(b)a= 6(c)a=13(d)a= mathcentre 20093. For each of the following pairs of functions , state whether the graphs are related by a reflectionin thex-axis, a reflection in thef(x)-axis, a reflection in the linef(x) =x, a reflection in thelinef(x) = x, or that the graphs are not related by any of these reflections.(a)f(x) = 3xandf(x) =(13)x(b)f(x) = log6xandf(x) = 6x(c)f(x) = log6xandf(x) =(16)x(d)f(x) = log1/3xandf(x) = log3x(e)f(x) =(13)xandf(x) =(16) (x)xf(x) = ( )x13f(x) = ( )x16f(x) = 3xf(x) = 6xf(x) = (x)xf(x) = log1/3 xf(x) = log1/6 xf(x) = log3 xf(x) = log6 mathcentre 20093.(a) Reflect in thef(x)-axis(b) Reflect in the linef(x) =x(c) Not related by any of these reflections(d) Reflect in thex-axis(e) Not related by any of these mathcentre 2009


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