Mathematics (2 unit) – Exponential and Logarithmic Functions

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Exponential and Logarithmic Functions

We shall now investigate the exponential and logarithmic functions. Firstly, we shall study and revise what should now be the very familiar index laws. We shall then study the exponential function due to the familiarity of the function, after which we shall develop the mathematical frame work for the logarithmic function.

TheIndex Laws

The index notation is used to simplify what would normally be very long notation. The expression

Latex formula

Has the value of Latex formula multiplyed by it self Latex formula times. That is,

Latex formula

Latex formula times

Recall the four main index laws,

 Latex formula
Latex formula
Latex formula
Latex formula

These should be familiar. From these four laws we may extrapolate a few extra rules (or theorems) to ease our mathematical work.

 Latex formula
Latex formula
Latex formula
Latex formula

 

Example 1

Simplify the following using the index laws:

a) Latex formula

 

b) Latex formula

 

Solution 1

a) Latex formula

 

b) Latex formula

 

 

Example 2

Find the exact value of Latex formula if

Latex formula

Solution 2

We simply have to substitute the values into the expressions and use the index laws to simplify the expression. But firstly, note that,

Latex formula

Latex formula

Latex formula

Now, substituting these into the expression gives,

Latex formula

Latex formula

Now multiplying top and bottom through by Latex formula and dividing top and bottom by Latex formula gives,

 

Latex formula

The Exponential Function

The most general form of the exponential function is Latex formula where Latex formula and Latex formula is a constant. We shall now investigate the graphs of exponential functions.

For the general function Latex formula the graph of the function has a general shape given below. But firstly we should consider asymptotic behaviour and any possible intercepts. As Latex formula. Also, Latex formula and there does not exist Latex formula such that Latex formula. Also, the domain of the function is all real Latex formula and the range is all positive real values of Latex formula. Hence we can obtain the graph,

graphs of exponential functions

Observe the steepness of the curve for Latex formula. This is due to the rapid growth of the exponential function.

Now, the graph of Latex formula is similar, but of course is a reflection of the graph above about the Latex formula-axis resulting in the graph;

reflection of the graph

Now by using similar techniques to those used in drawing Latex formula, one can draw any variation of these exponential functions with ease. We shall consider some examples to illustrate the notions here.

Example 3

Sketch the graphs of the following functions;

a) Latex formula

 

b) Latex formula

 

c) Latex formula

Solution 3

a) To sketch this we simply consider the graph of Latex formula and simply shift this one unit up. That is we have the graph,

shifting of the graph

 

b) Here we see that this graph is exactly the same as that of Latex formula except for the shifting of the graph Latex formula unit to the left. (One can easily see this by considering the point at which Latex formula, in this case it is the point Latex formula indicating a shift of the graph one unit to the left.) An important point is to find the Latex formula-intercept. At Latex formula, Latex formula, and hence the Latex formula-intercept is Latex formula.

shifting of the graph two

c) To sketch this curve, we firstly note that as Latex formula and as Latex formula. Now at Latex formula, Latex formula and hence the Latex formula intercept of this graph is Latex formula. The Latex formula-intercept of the graph is at Latex formula, which of course at the moment does not have any meaning, but will once we study the logarithmic function in the sections ahead. Putting this information together on a diagram gives;

intercept of the graph

Differential calculus of the exponential function

The differential calculus of the exponential function is quite straightforward. However, before delving into the calculus aspect, a new number must be introduced. The natural base Latex formula, is an irrational number approximately equal to Latex formula. This number is the most important and frequently used base in mathematics. The reason why this number is so incredibly important is obvious in the below formulae.

So, to illustrate the importance of Latex formula we consider the derivative of Latex formula.

 Latex formula

This function has the remarkable property that its derivative is equal to itself! This is why Latex formula is so important and the reason why almost all calculus involving the exponential functions and logarithmic functions will be done in the base Latex formula. Now, to generalise the above rule we have that,

Latex formulaLatex formula

We shall study the differentiation process of functions with bases other than Latex formula in a later chapter.

We shall now study some examples to illustrate the process of finding derivatives. Note that all the previous differentiation rules (product rule, quotient rule, chain rule) all apply.

Example 4

Differentiate the following functions with respect to Latex formula.

a) Latex formula

 

b) Latex formula

 

c) Latex formula

 

d) Latex formula

 

e) Latex formula

 

Solution 4

a) Latex formula Latex formula

 

b) Latex formula Latex formula

 

c) Latex formula Latex formula

 

d) Latex formula Latex formula Latex formula Latex formula Latex formula Latex formula Latex formula. Of course you could have considered and applied the product rule onto this function.

 

e) Latex formula Latex formula Latex formula

The process of finding tangents, normals, minima, maxima and points of inflexion all still apply as before. You must be able to use the derivative effectively to obtain the required results throughout the questions. Consider the below examples which illustrate the use of the derivative.

Example 5

Find the tangent at Latex formula, on the curve Latex formula.

Solution 5

So, to find the tangent, we simply differentiate with respect to Latex formula to find the gradient at Latex formula, and then use the point gradient formula to find the equation o the tangent.

Latex formulaLatex formula

We have that,

Latex formulaLatex formula

And that

Latex formula

Using the point gradient formula,

Latex formula

Hence we have that the equation of the tangent is,

Latex formula

 

Example 6

Find the stationary points of the curve Latex formula

Solution 6

Differentiating with respect to Latex formula gives,

Latex formula

Now, setting Latex formula to zero and solving the resultant equation gives,

Latex formula

Latex formula

Hence we have that there is a stationary point at Latex formula. Now differentiating with respect to Latex formula again gives,

Latex formula

 

Now at Latex formula, Latex formula Latex formula and hence Latex formula is a minimum stationary point on the curve Latex formula.

Integral Calculus of the Exponential Function

Since integration is the exact reverse process of differentiation, then it follows that we can integrate exponential functions according to the following rules;

 Latex formula
Latex formula

And in general,

Latex formula

We shall now look at some examples illustrating the types of questions involved. Note that all the previous taught rules relating to integration still apply, and that the new taught rules only apply to finding the anti-derivative of the exponential function.

Example 7

Find

Latex formula

Solution 7

We may use the second rule presented above.

Latex formula

 

Example 8

Find the indefinite integral;

Latex formula

Solution 8

To find this integral, we use the third rule, with Latex formula. Since we have that Latex formulaLatex formula, then we must place the coefficient of Latex formula and balance by multiplying the entire expression by Latex formula to integrate. That is,

Latex formula

Latex formula

 

After applying the integration rule.

 

Example 9

Find the value of

 Latex formula

Solution 9

Firstly, we note that we may use the second of the rules shown above to find the primitive of the function.

Latex formula

Latex formula

Note: You must take care when rearranging the Latex formulaLatex formula expression to obtain an integral involving exponential functions. You can only rearrange constants, to obtain the Latex formulaLatex formula that is required.

Logarithms

A logarithm is another name for an index. It tells us how one number can be written as a power of another number. Mathematically, if we have that

Latex formula

Then it follows that Latex formula is the logarithm of Latex formula base Latex formula. To understand the word logarithm replace the word logarithm by index in the previous sentence. The notation for the logarithm of Latex formula base Latex formula we use is Latex formula. Hence we have that if;

Latex formula

Note: Your calculator has two buttons for calculating logarithm; log and ln. The first button calculates logarithms to the base ten, which were important before the advent of modern calculators, but aren’t used very frequently now. The second button is more commonly used and is the logarithm in base . The importance of this logarithm relates to the calculus of the logarithmic functions.

Note: The notation used for logarithms at times becomes quite confusing. Throughout this text and most othersLatex formula, will indicate the natural logarithm (logarithm base Latex formula ), Latex formula will also indicate the natural logarithm, and Latex formula will indicate the logarithm taken with the base ten.

The Logarithm Laws

As there are index laws for exponential functions, there also exist logarithm laws for logarithmic functions. These laws are very similar to the index laws. They are listed below, and must be committed to memory.

 Latex formula
Latex formula
Latex formula
Latex formula
Latex formula

These laws may be proven using the index laws, however we shall leave those to the discretion of the student. Consider the below examples to illustrate the use of these laws.

Example 10

Write the following as the logarithm of a single expression or number.

a) Latex formula

 

b) Latex formula

 

c) Latex formula

 

d) Latex formula

 

Solution 10

a) Latex formula

 

b) Latex formula

 

c) Here, we express each logarithm in terms of the same number. That is,

 

Latex formula

Latex formula

 

After collecting like terms we have

Latex formula

 

d) Here we express each logarithm as a logarithm of a power of 2, after which we use the power rule to bring to power of the two down the front of the logarithm.

Latex formula

Latex formula

There are three extra theorems which are of extreme importance when dealing with the logarithmic function. We shall not look at examples involving these, however it is of the utmost importance that students remember them.

Change of Base:

Latex formula

Inverse Function rule:

Latex formula

The Logarithmic Function

The logarithmic function is of the form Latex formula where Latex formula is a constant greater that Latex formula. The most commonly encountered logarithmic function is Latex formula. Before we investigate the graphs, it should be noted that the domain of the function is Latex formula and the range of the function is all the real values of Latex formula. Students may realise that the domain of the logarithmic function is equal to the range of the exponential and the range of the logarithmic function is equal to the domain of the exponential. This is no coincidence, and is due to the fact that the logarithmic function is the inverse of the exponential, and hence the two graphs (that is, the graph of the exponential and the graph of the logarithmic function) are reflections of one another about the line Latex formula. A graph of this function is shown below.

logarithmic function graph

It can be seen that as Latex formula, Latex formula, and that as Latex formula, albeit rather slowly. The rate of growth of the logarithmic function is very slow, and attributed to the value of its derivative. This is why the function seems to flatten out but in fact is still growing indefinitely, just at a rate that is quite slow.

The below graph is a graph of the function Latex formula.

logarithmic function fhift graph

It can be seen that all other forms of the logarithmic function may be graphed by some form of a shift, stretch or flip.

Note: Students commonly lose marks by not making their logarithmic graphs taper off as Latex formula gets large. To ensure that maximum marks are obtained, ensure that the graph is growing at a very slow rate as Latex formula.

Example 11

Sketch the graph of Latex formula

 

Solution 11

Now, to sketch this curve we firstly use the logarithm laws to simplify the analysis required.

 

Latex formula

 

Now, since at Latex formula, the argument of the logarithm is 0, then it follows that there exits an asymptote at Latex formula. Also, since for Latex formula, the value of Latex formula is negative, then it follows that the function is only defined for Latex formula, with an asymptote at Latex formula.

The factor of Latex formula in front of the logarithm is a “stretching” factor and will stretch the curve.

As always, we find the Latex formula and Latex formula intercepts. For the Latex formula-intercept, we have that at Latex formula, Latex formula. Hence we have that the Latex formula-intercept is Latex formula. Now, to find the Latex formula-intercepts, we simply look at the solutions of Latex formula.

Latex formula

Taking the exponential of both sides gives,

Latex formula

Now, recall the rule,

Latex formula

This rule is applicable here, since Latex formula.

Thus we have here,

Latex formula

Hence the Latex formula-intercept is Latex formula.

Now, using this information and drawing the curve gives,

exponential of both sides

The Differential Calculus of the Logarithmic Function

The derivative of the Logarithmic function solves a very tedious problem formed when calculus was developed. What function has derivative equal to Latex formula? This question turns out to be answered by the function Latex formula. The derivation of the differential will not be shown, instead we shall simply show the derivatives and students should remember these results.

 Latex formula

and in general we have that;

 Latex formula

Notice here that we have only shown the rule for differentiation of logarithmic functions taken with base Latex formula. Other bases will be dealt with in a later section.

Consider the below examples which illustrate differentiation of the logarithmic function. Note that the previous rules obtained, still apply in this case (product rules, quotient rules and chain rules).

Example 12

Differentiate the following functions:

a) Latex formula

 

b) Latex formula

 

c) Latex formula

 

d) Latex formula

 

e) Latex formula

Solution 12

a) Latex formula

 

b) Latex formula

 

c) Latex formula

 

d) Latex formula

 

e) Latex formula

We shall now look at some exam style questions involving the differential calculus. Note that all previous methods apply to finding tangents, normals, maxima and minima.

Example 13

Find the equation of the tangent to the curve Latex formula at Latex formula.

Solution 13

At Latex formula we have that Latex formula. Hence the point required is Latex formula. Now, to find the gradient we differentiate with respect to Latex formula to find the gradient at Latex formula.

 

Latex formula

 

At Latex formula we have that,

 

Latex formula

 

Now, using the point-gradient gives,

 

Latex formula

 

Hence we have that,

 

Latex formula

 

Is the tangent at Latex formula.

 

Example 14

Find the stationary point on the curve Latex formula and determine its nature.

Solution 14

We firstly differentiate the curve and set Latex formula, in which case we can solve the resultant equation for Latex formula which gives the stationary points.

 

Latex formula

 

We have that,

 

Latex formula

 

Hence we have that,

 

Latex formula(However, Latex formula as Latex formula is not defined at this point)

 

Latex formula

 

Now solving the second equation gives,

 

Latex formula

 

Hence we have a stationary point at Latex formula. We shall use the second derivative test to find the nature of the stationary point.

 

Latex formula

 

Hence we have that the stationary point at Latex formula is a minimum stationary point.

The Integral Calculus of the Logarithmic Function

By use of the fundamental theorem of calculus we have the results;

Latex formula

and the more general result;

 Latex formula

We shall illustrate use of these results in the example below.

Example 15

Find

Latex formula

Solution 15

Firstly we identify that Latex formula. Hence we have that Latex formulaLatex formula, and we must rearrange the coefficients to obtain a Latex formula on the numerator of the coefficient.

 

Latex formula

Latex formula

 

Upon using the appropriate integration formula.

 

Example 16

Find

Latex formula

Solution 16

Now, in this case we have that Latex formulaLatex formula. Hence we have to adjust the coefficient of Latex formula to obtain the expression Latex formulaLatex formula on the numerator of the rational function.

 

Latex formula

 

Latex formula

 

 

Example 17

Find

Latex formula

Solution 17

Here we cannot use the logarithmic integration rule, since Latex formulaLatex formula. As there does not exist an Latex formula in the numerator, then we have that we cannot use the logarithm laws. Instead this is an integral of a power of Latex formula and we shall treat it as thus;

 

Latex formula

Latex formula

Latex formula

We shall now consider some exam style questions involving integration. Note that all previous formulae still apply for the logarithmic function as well.

Example 18

Find the area bound by the curve Latex formula, the Latex formula-axis and the lines Latex formula and Latex formula.

Solution 18

Firstly we draw a diagram of the curve. Notice that there is a vertical asymptote at Latex formula and a horizontal asymptote at Latex formula. Hence we have the curve,

diagram of the curve

So, to calculate the area, we simply have to consider

 

Latex formula

Latex formula

Latex formula

Latex formula

 

Example 19

Find the derivative of Latex formula with respect to Latex formula. Hence find

Latex formula

Solution 19

Firstly, we differentiate the expression with respect to Latex formula.

 

Latex formula

Latex formula

 

Now, to find the integral required, we integrate both sides with respect to Latex formula. That is,

 

Latex formula

 

Now, the LHS of the expression is simply equal to Latex formula by the fundamental theorem of calculus which states that integration and differentiation are inverse (reverse) processes.

 

Latex formula

 

Now, separating the integral on the RHS into two separate integrals gives,

 

Latex formula

 

Now, we have that,

 

Latex formula

 

Now, rearranging terms gives,

 

Latex formula

 

Now, replacing Latex formula by Latex formula gives,

 

Latex formula

Differentiation and Integration of Logarithms and Exponentials of Bases other than e

To differentiate and integrate expressions involving other bases, we simply use the exponential inverse rule, or the change of base rule that was presented before. These rules allow for the base to be changed to that of Latex formula and hence allow for the integral or differential to be evaluated. That is,

 Latex formula

And we have that,

 Latex formula

Using these rules, we are then able to differentiate and integrate these functions.

Example 20

Differentiate with respect to Latex formula;

a) Latex formula

 

b) Latex formula

Solution 20

a)

Latex formula

 

Latex formula

 

Since the Latex formula is a constant. Now converting back gives,

 

Latex formula

 

b)

Here we first change the base,

 

Latex formula

 

Now, differentiating the expression with respect to Latex formula gives,

 

Latex formula

Latex formula

Latex formula

Latex formula

 

Example 21

Find the anti-derivative of Latex formula.

Solution 21

So we are required to find;

 

Latex formula

 

Now, performing the change of base gives,

 

Latex formula

 

Recall that the derivative of the expression in the index must be out the front in order to be able to integrate.

 

Latex formula

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