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
Contents
 1 TheIndex Laws
 2 The Exponential Function
 3 Differential calculus of the exponential function
 4 Integral Calculus of the Exponential Function
 5 Logarithms
 6 The Logarithm Laws
 7 The Logarithmic Function
 8 The Differential Calculus of the Logarithmic Function
 9 The Integral Calculus of the Logarithmic Function
 10 Differentiation and Integration of Logarithms and Exponentials of Bases other than e
The index notation is used to simplify what would normally be very long notation. The expression
Has the value of multiplyed by it self times. That is,
times 
Recall the four main index laws,

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

Example 1Simplify the following using the index laws: a)
b)
Solution 1a)
b)

Example 2Find the exact value of if Solution 2We simply have to substitute the values into the expressions and use the index laws to simplify the expression. But firstly, note that, Now, substituting these into the expression gives, Now multiplying top and bottom through by and dividing top and bottom by gives,

The Exponential Function
The most general form of the exponential function is where and is a constant. We shall now investigate the graphs of exponential functions.
For the general function the graph of the function has a general shape given below. But firstly we should consider asymptotic behaviour and any possible intercepts. As . Also, and there does not exist such that . Also, the domain of the function is all real and the range is all positive real values of . Hence we can obtain the graph,
Observe the steepness of the curve for . This is due to the rapid growth of the exponential function.
Now, the graph of is similar, but of course is a reflection of the graph above about the axis resulting in the graph;
Now by using similar techniques to those used in drawing , one can draw any variation of these exponential functions with ease. We shall consider some examples to illustrate the notions here.
Example 3Sketch the graphs of the following functions; a)
b)
c) Solution 3a) To sketch this we simply consider the graph of and simply shift this one unit up. That is we have the graph,
b) Here we see that this graph is exactly the same as that of except for the shifting of the graph unit to the left. (One can easily see this by considering the point at which , in this case it is the point indicating a shift of the graph one unit to the left.) An important point is to find the intercept. At , , and hence the intercept is . c) To sketch this curve, we firstly note that as and as . Now at , and hence the intercept of this graph is . The intercept of the graph is at , 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; 
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 , is an irrational number approximately equal to . 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 we consider the derivative of .
This function has the remarkable property that its derivative is equal to itself! This is why is so important and the reason why almost all calculus involving the exponential functions and logarithmic functions will be done in the base . Now, to generalise the above rule we have that,
‘ 
We shall study the differentiation process of functions with bases other than 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 4Differentiate the following functions with respect to . a)
b)
c)
d)
e)
Solution 4a)
b)
c)
d) . Of course you could have considered and applied the product rule onto this function.
e) 
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 5Find the tangent at , on the curve . Solution 5So, to find the tangent, we simply differentiate with respect to to find the gradient at , and then use the point gradient formula to find the equation o the tangent. ‘ We have that, ‘ And that Using the point gradient formula, Hence we have that the equation of the tangent is, 
Example 6Find the stationary points of the curve Solution 6Differentiating with respect to gives, Now, setting to zero and solving the resultant equation gives, Hence we have that there is a stationary point at . Now differentiating with respect to again gives,
Now at , and hence is a minimum stationary point on the curve . 
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;

And in general,
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 antiderivative of the exponential function.
Example 7Find Solution 7We may use the second rule presented above. 
Example 8Find the indefinite integral; Solution 8To find this integral, we use the third rule, with . Since we have that ‘, then we must place the coefficient of and balance by multiplying the entire expression by to integrate. That is,
After applying the integration rule. 
Example 9Find the value of
Solution 9Firstly, we note that we may use the second of the rules shown above to find the primitive of the function. 
Note: You must take care when rearranging the ‘ expression to obtain an integral involving exponential functions. You can only rearrange constants, to obtain the ‘ 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
Then it follows that is the logarithm of base . To understand the word logarithm replace the word logarithm by index in the previous sentence. The notation for the logarithm of base we use is . Hence we have that if;
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 others, will indicate the natural logarithm (logarithm base ), will also indicate the natural logarithm, and 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.

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 10Write the following as the logarithm of a single expression or number. a)
b)
c)
d)
Solution 10a)
b)
c) Here, we express each logarithm in terms of the same number. That is,
After collecting like terms we have
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. 
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: 
Inverse Function rule:
The Logarithmic Function
The logarithmic function is of the form where is a constant greater that . The most commonly encountered logarithmic function is . Before we investigate the graphs, it should be noted that the domain of the function is and the range of the function is all the real values of . 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 . A graph of this function is shown below.
It can be seen that as , , and that as , 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 .
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 gets large. To ensure that maximum marks are obtained, ensure that the graph is growing at a very slow rate as .
Example 11Sketch the graph of
Solution 11Now, to sketch this curve we firstly use the logarithm laws to simplify the analysis required.
Now, since at , the argument of the logarithm is 0, then it follows that there exits an asymptote at . Also, since for , the value of is negative, then it follows that the function is only defined for , with an asymptote at . The factor of in front of the logarithm is a “stretching” factor and will stretch the curve. As always, we find the and intercepts. For the intercept, we have that at , . Hence we have that the intercept is . Now, to find the intercepts, we simply look at the solutions of . Taking the exponential of both sides gives, Now, recall the rule, This rule is applicable here, since . Thus we have here, Hence the intercept is . Now, using this information and drawing the curve gives, 
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 ? This question turns out to be answered by the function . The derivation of the differential will not be shown, instead we shall simply show the derivatives and students should remember these results.
and in general we have that;
Notice here that we have only shown the rule for differentiation of logarithmic functions taken with base . 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 12Differentiate the following functions: a)
b)
c)
d)
e) Solution 12a)
b)
c)
d)
e) 
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 13Find the equation of the tangent to the curve at . Solution 13

Example 14Find the stationary point on the curve and determine its nature. Solution 14We firstly differentiate the curve and set , in which case we can solve the resultant equation for which gives the stationary points.
We have that,
Hence we have that,
(However, as is not defined at this point)
Now solving the second equation gives,
Hence we have a stationary point at . We shall use the second derivative test to find the nature of the stationary point.
Hence we have that the stationary point at is a minimum stationary point. 
The Integral Calculus of the Logarithmic Function
By use of the fundamental theorem of calculus we have the results;
and the more general result;
We shall illustrate use of these results in the example below.
Example 15Find Solution 15Firstly we identify that . Hence we have that ‘, and we must rearrange the coefficients to obtain a on the numerator of the coefficient.
Upon using the appropriate integration formula. 
Example 16Find Solution 16Now, in this case we have that ‘. Hence we have to adjust the coefficient of to obtain the expression ‘ on the numerator of the rational function.

Example 17Find Solution 17Here we cannot use the logarithmic integration rule, since ‘. As there does not exist an in the numerator, then we have that we cannot use the logarithm laws. Instead this is an integral of a power of and we shall treat it as thus;

We shall now consider some exam style questions involving integration. Note that all previous formulae still apply for the logarithmic function as well.
Example 18Find the area bound by the curve , the axis and the lines and . Solution 18Firstly we draw a diagram of the curve. Notice that there is a vertical asymptote at and a horizontal asymptote at . Hence we have the curve, So, to calculate the area, we simply have to consider

Example 19Find the derivative of with respect to . Hence find Solution 19Firstly, we differentiate the expression with respect to .
Now, to find the integral required, we integrate both sides with respect to . That is,
Now, the LHS of the expression is simply equal to by the fundamental theorem of calculus which states that integration and differentiation are inverse (reverse) processes.
Now, separating the integral on the RHS into two separate integrals gives,
Now, we have that,
Now, rearranging terms gives,
Now, replacing by gives,

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 and hence allow for the integral or differential to be evaluated. That is,
And we have that,
Using these rules, we are then able to differentiate and integrate these functions.
Example 20Differentiate with respect to ; a)
b) Solution 20a)
Since the is a constant. Now converting back gives,
b) Here we first change the base,
Now, differentiating the expression with respect to gives,

Example 21Find the antiderivative of . Solution 21So we are required to find;
Now, performing the change of base gives,
Recall that the derivative of the expression in the index must be out the front in order to be able to integrate.
