exponential_logarithmic_functions.pdf

white Exponential functionsLogarithmic functions

Linear formsCurve sketching

Lecture

Ong Ming Tze

International Medical University

May 15, 2014

Ong Ming Tze Lecture



1 Exponential functions

2 Logarithmic functions

3 Linear forms

4 Curve sketching

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What are exponential functions

In general exponential functions are functions of theform

f (x) = ax (1)

Note that in (1), a must be a postive number and also notequal to 1.

x is a variable and can be any real number. So for example g(x) = 3.2x and h(x) = 0.32x are

exponential functions. On the other hand k(x) = 1x is NOT an exponential

function since k(x) is just the constant 1.

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In Figure 1a below we see how an exponential functionwould look like if a is a number greater than 0 and lessthan 1.

In Figure 1c below we see how an exponential functionwould look like if a is a number greater than 1.

Figure 1:

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Note that y = 1x in Figure 1b above is NOT anexponential function.

What we should observe from Figure 1 is that given somefunction

f (x) = ax

if 0 < a< 1 then ax is a decreasing function, and that ifa> 1, then ax is an increasing function.

Hopefully you remember what increasing and decreasingfunctions are.

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In figure 2 below we give more examples of graphs ofexponential functions.

Figure 2:

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Observations

Notice in figure 2 above that the graphs of y = 2x andy = (12)

x = 2x are reflections of each other about they -axis,

and that, the the graphs of y = 4x and y = (14)x = 4x arealso reflections of each other about the y -axis.

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Rules for exponential functions

If a,b > 0 and x ,y R, then1. axay = ax+y 2. a

x

ay = axy 3. (ax)y = axy 4. (ab)x = axbx

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The irrational number e

Recall that an exponential function is a function of theform

f (x) = ax (2)

where a is a constant greater than 0 and not equal to 1. The constant a in (2) is known as the base of the

exponential function f (x). One frequently occuring base of exponential functions is

the irrational number denoted by e.

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Definition of the number ee is the number lim

n(1+xn )

n

What you should take note of is that the exponentialfunction ex occurs commonly.

To around 10 decimal places, e 2.7182818284. The function

f (x) = ex (3)

is known as the natural exponential function .

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Figure 3 below shows the graph of y = ex .

Figure 3:

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From Figure 3 above note the following,The domain and range of the function f(x) = ex

1 The domain of ex is the entire set of real numbers R.2 The range of ex is the set of positive numbers (0,).

Observe from the graph of Figure 3 that ex is never 0. That is to say the graph of y = ex never touches or

crosses the x-axis.

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The natural exponential function ex

What is so special about the natural exponentialfunction ex?

Figure 4:Lecture



Figure 4 above shows that the gradient of the graph ofy = ex at points on the graph are nothing but the valuesof the y -coordinates of the points.

In terms of calculus this means thatddx{ex}= ex (4)

We will return to (4) later on. But for now please rememberthis important result.

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Logarithmic functions

If a> 0 and a 6= 1, then the exponential function f (x) = ax is either increasing or

decreasing. Why? This means that f (x) has an inverse function f1 known

as the logarithmic function with base a and is denotedby

loga (5)

So f1(x) = loga x .

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So the very important relationship between exponential andlogarithmic functions is that they are inverse functions of eachother.

So if a> 0 and a 6= 1 then, (as you have seen before)

y = ax loga y = x (6)

In the figure below we see how the two exponential andlogarithmic functions look like.

Remember because they are inverses of each other thetwo curves are reflections of each other about the linewith equation y = x .

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Observe the properties of the exponential andlogarithmic function

Figure 5:

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Important properties of the logarithmic function

If a> 1 What are the domains of ax and loga x? What are the values that the functions ax and loga x take?

As we have seen previously, if a > 0, x ,y > 0 and r is any realnumber, then

1 loga x+ loga y = loga (xy)2 loga x loga = loga ( xy )3 loga x

r = r loga x

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Graphs of logarithmic functions with different bases

Figure 6:

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Natural logarithms

Of all the possible bases a for the logarithmic function loga x ,the most convenient choice for the base a is the number ewhich we have seen earlier. The logarithmic function with base e is written as ln x . In other words,

ln x loge x (7)

We thus have the following

y = ex ln y = x

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Example 1: Solve the equation e53x = 10.

Example 2: Express ln a+ 12 ln b using a single logarithm.

Example 3: Solve the equation ln(1+x) = 2.

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Example 4: A cup of tea, at temperature T C cools downaccording to the formula:

T = 80ekt +15

where t is the time in seconds after the cup of tea was made.

1 What is the initial temperature of the tea?2 The tea cools by 20C after the first minute. Find the value

of k .3 How long will it take for the temperature to drop to 50C.

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Remember that earlier we have seen the rule regardingthe changing of bases of logarithms

loga b logc blogc a

(8)

where a,b and c > 0. It follows then, that,

loga x loge bloge a

ln xln a

. (9)

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Sometimes relationsips between two variables can bemodelled with exponential functions.

These relationships however can be reduced to linear formso that the constants in the relationship can be calculated.

Suppose we have a relationship of the form

y = axn (10)

where a is a constant.

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Then taking logs (natural base e) on both sides of equation(1) we get

ln y = ln axn

ln y = ln a+ ln xn ln y = n (lnx)+ ln a (11)

Comparing equation (11) with the form Y =mX +C, wecan see that plotting values of ln y against values of ln xgives a straight line whose gradient is n and whoseintercept on the vertical axis is ln a.

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Example 5: Two variables p and q are related by the equationp = aqn. Some values of p and q are found from an experiment.The graph in figure 7 shows the result of plotting ln p againstln q. Estimate the values of a and n.

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Figure 7:

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Graphs of functions

Given the graphs of basic functions we can obtain thegraphs of related functions.

For example in figure 8 below, given the graph of y = f (x)we can see that the graph of y = f (xa), where a> 0 isnothing but the graph of y = f (x) shifted a units to theright.

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Similarly in figure 8 below observe that given the graph ofy = f (x) we can see that the graph of y = f (x)a, wherea> 0 is nothing but the graph of y = f (x) shifted a unitsdownwards.

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Translations of the graph of y = f (x)

Figure 8:

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Reflections of the graph of y = f (x) about the x andy -axes.

Figure 9:

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Example : Sketch the graph of1 y = 3x

2 y = 3x

3 y =3x4 y = 23x

Lecture

Exponential functionsLogarithmic functionsLinear formsCurve sketching

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exponential_logarithmic_functions.pdf