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Logarithmic and exponential functions

Module by: Sunil Kumar Singh

Logarithmic and exponential functions are closely related functions. Logarithmic functions are useful in interpreting expressions/ equations, in which exponents are unknown. On the other hand, exponential functions are representation of natural process or mathematical relations, having exponential growth or decay. We shall encounter many expressions, involving these two functions in mathematics, while analyzing or describing processes.
The two functions have simple derivatives (we shall learn about derivatives in differential calculus). This, in turn, is helpful in carrying out integration, involving these functions.
Symbolically, logarithmic and exponential functions, respectively, are :
f x = log a x f x = log a x
and
f x = a x f x = a x
In this module, we shall find out that logarithmic and exponential functions, as a matter of fact, are inverse functions to each other.

Logarithmic functions

A logarithmic function gives “exponent” of an expression in terms of a base, “a”, and a number, “x”. The following two representations, in this context, are equivalent :
a y = x a y = x
and
f x = y = log a x f x = y = log a x
where :
  1. The base “a” is positive real number, but excluding “1”. Symbolically, a > 0, a ≠ 1 a > 0, a ≠ 1 .
  2. The “x” represents result of exponentiation, “ a y a y ” and is also a positive real number. Symbolically, x >0.
  3. The exponent “y” i.e. logarithm of “x” is a real number.
Note that neither “a” nor “x” equals to zero.
The equation of a logarithmic value for “x” on a certain base represents logarithmic function. In words, we can say that a logarithmic function associates every positive real number (x) to a real valued exponent (y), following the relation symbolically represented as :
f x = log a x ; a , x > 0, a ≠ 1 f x = log a x ; a , x > 0, a ≠ 1
We exclude "a = 1" as logarithmic function is not relevant to this base. Consider the exponentiation :
1 y = 1 1 y = 1
We can easily see here that whatever be the exponent, the value of logarithmic function is “1". Hence, base “1” is irrelevant as exponent “y” is not uniquely associated with “x”.
We can associate logarithmic concept easily with decimal values represented as a power of base “10”. Consider the examples given here :
10 - 1 = 0.1 10 - 1 = 0.1
10 0 = 1 10 0 = 1
10 2 = 100 10 2 = 100
Note that base, in this case, is a positive number, “10”. The resulting value (x) is also a positive number, irrespective of whether exponent is positive or negative. However, exponents are zero or positive or negative integer. This is a specific example. We can easily visualize that exponents can be any real number.
From this discussion, we conclude that :
Value of “x” = Domain = 0, ∞ Value of “x” = Domain = 0, ∞
Value of “y” = Range = R Value of “y” = Range = R

Base

The base of the logarithmic function can be any positive number. However, “10” and “e” are two common bases that we often use. Here, “e” is a mathematical constant given by :
e = 2.718281828 e = 2.718281828
If we use “e” as the base, then the corresponding logarithmic function is called “natural” logarithmic function. The plots, here, show logarithmic functions for two bases (i) 10 and (ii) e.
Logarithmic function
le1.gif
Figure 1: The plots of logarithmic function on different bases.
Note that expanse of logarithmic function "f(x)" is along y – axis on either side of the x-axis, showing that its range is R. On the other hand, the expanse of “x” is limited to positive side of x-axis, showing that domain is positive real number. Further, irrespective of bases, all plots intersect x-axis at the same point i.e. x = 1 as :
x = a y = a 0 = 1 x = a y = a 0 = 1

Graphs

We have noted the importance of base “1” for logarithmic function. Base “1” plays an important role for determining nature of logarithmic function. Here, we have drawn plots for two cases : (i) a < 1 and (ii) a > 1.
If the base is greater than zero, but less than “1”, then the logarithm function asymptotes to positive y-axis.
Logarithmic function
le2.gif
Figure 2: The plot of logarithmic function, when base is less than "1".
If the base is greater than “1”, then the logarithm function asymptotes to negative y-axis.
Logarithmic function
le3.gif
Figure 3: The plot of logarithmic function, when base is greater than "1".

Logarithmic identities

Some of the important logarithmic identities are given here without proof. Idea, here, is to simply equip ourselves so that we can work with logarithmic functions in conjunction with itself or with other functions.
log a x y = log a x + log a y log a x y = log a x + log a y
log a x y = log a x - log a y log a x y = log a x - log a y
log a x y = y log a x log a x y = y log a x
log a x 1 y = log a x y log a x 1 y = log a x y

Change of base

Sometimes, we are required to work with logarithmic expressions of different bases. In such cases, we convert them to same base, using following relation :
log a x = log b x log b a log a x = log b x log b a

Example

Problem 1 : Find the domain of the function given by :
f x = 1 log 10 1 − x f x = 1 log 10 1 − x
Solution : The given function is reciprocal of a logarithmic function. Therefore, we first need to ensure that logarithmic function does not evaluate to zero for a value of “x”. A logarithmic function evaluates to zero if its argument is equal to “1”. Hence,
1 − x ≠ 1 1 − x ≠ 1
⇒ x ≠ 0 ⇒ x ≠ 0
Further domain of logarithmic function is a positive real number. For that :
1 - x > 0 1 - x > 0
⇒ 1 > x ⇒ 1 > x
⇒ x < 1 ⇒ x < 1
Combining two results, the domain of the given function is :
Domain of “f” = - ∞ , 1 − { 0 } Domain of “f” = - ∞ , 1 − { 0 }

Exponential function

An exponential function relates every real number “x” to the exponentiation, “ a x a x ”. In other words, we can say that an exponential function associates every real number (x) to a function given by :
f x = y = a x f x = y = a x
where :
  1. The base “a” is positive real number, but excluding “1”. Symbolically, a > 0, a ≠ 1 a > 0, a ≠ 1 .
  2. The exponent “x” is a real number.
  3. The “y” represents the result of exponentiation, “ a x a x ” and is a positive real number. Symbolically, y >0.
Note that neither “a” nor “y” equals to zero.
It can be easily seen that roles of “x” and “f(x)” have exchanged here with respect to logarithmic function. Here, “x” is the exponent i.e. the logarithmic value and “f(x)” is the result of exponentiation. For this reason, we say that logarithmic and exponential functions are inverse to each other.
In this case,
Value of “x” = Domain = R Value of “x” = Domain = R
Value of “y” = Range = 0, ∞ Value of “y” = Range = 0, ∞
As expected, we see that domain and range of logarithmic and exponential functions have also been exchanged.
The exponential function corresponding to base “e” is called “natural” exponential function. The plots of exponential functions for two cases (i) a < 1 and (ii) a > 1 are considered here as in the case of logarithmic function.
If the base is greater than zero, but less than “1”, then the exponential function asymptotes to positive x-axis.
Exponential function
le4.gif
Figure 4: The plot of exponential function, when base is less than "1".
If the base is greater than “1”, then the logarithm function asymptotes to negative x-axis.
Exponential function
le5.gif
Figure 5: The plot of exponential function, when base is greater than "1".
Note that expanse of logarithmic function is along x – axis on either side of the y-axis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of y-axis, showing that its range is positive real number. Further, irrespective of bases, all plots intersect y-axis at the same point i.e. y = 1 as :
y = a x = a 0 = 1 y = a x = a 0 = 1

Example

Problem 2 : Find all real values of “x” such that :
1 - e 1 x − 1 > 0 1 - e 1 x − 1 > 0
Solution Solving for the exponential function, we get following inequality :
⇒ e 1 x − 1 < 1 ⇒ e 1 x − 1 < 1
Now, we know that domain of an exponential function is “R”. However, this information is not helpful here to find values of “x” that satisfies the inequality. Taking natural logarithm on either side of the equation,
⇒ 1 x − 1 < log e 1 ⇒ 1 x − 1 < log e 1
But, logarithm of “1” i.e. log e 1 log e 1 is zero. Hence,
⇒ 1 x − 1 < 0 ⇒ 1 x − 1 < 0
Here, we should not solve for "x" to obtain the interval as it is in the rational polynomial form. The important point is to understand that we are required to find the values of "x" for which the given inequality holds.
⇒ 1 − x x < 0 ⇒ 1 − x x < 0
Now, we proceed to find the values of “x” for which left hand side expression is negative. Following sign convention, the roots of the numerator and denominator are “0” and “1”. For a value x = 2, the expression evaluates to a negative value. Hence, interval x > 1 gives negative value for the expression. Now, “1” appears only 1 time i.e. odd time. Therefore, the interval to the left of “1” gives positive value of the expression. Also, “0” appears only 1 time. The interval to the left of “0” is negative for the expression.
Sign convention
le6.gif
Figure 6: Positive and negative intervals.
Picking the intervals for which expression is negative :
Intervals
le7.gif
Figure 7: Intervals for which given function is positive.
- ∞ , 0 ∪ 1, ∞ - ∞ , 0 ∪ 1, ∞

Logarithmic inequality

The nature of logarithmic function is dependent on the base value. We know that base of a logarithmic function is a positive number excluding “1”. The value of “1” plays important role in deciding nature of logarithmic function and hence that of inequality associated to it. Let us consider an equality :
log a x > y log a x > y
What should we conclude : x > a y or x < a y x > a y or x < a y ? It depends on the value of “a”. We can understand the same by considering LHS of the inequality equal to an exponent z :
log a x = z log a x = z
If a > 1, then " a z a z " will yield a greater “x” than " a y a y ", because z>y (it is given by the inequality). On the other hand, if 0<a<1, then " a z a z " will yield a smaller “x” than " a y a y ", because z>y. We can understand this conclusion with the help of an example. Let
log 2 x > 3 log 2 x > 3
log 2 x = 4 log 2 x = 4
Clearly, 2 4 > 2 3 as 16 > 8 2 4 > 2 3 as 16 > 8
Let us now consider a <1,
log 0.5 x > 3 log 0.5 x > 3
log 0.5 x = 4 log 0.5 x = 4
Clearly, 0.5 4 < 0.5 3 as 0.0625 < 0.125 0.5 4 < 0.5 3 as 0.0625 < 0.125 . Thus, we finally conclude :
  1. log a x > y ⇔ x > a y ; a > 1 log a x > y ⇔ x > a y ; a > 1
  2. log a x > y ⇔ x < a y ; a < 1 log a x > y ⇔ x < a y ; a < 1
We have used two ways notation to indicate that interpretation of logarithmic inequality is true in either direction.

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