Logarithmic Functions

Sub Topics
The function of an expression in which the variable is a logarithm to any base is called a logarithmic function.

A logarithmic function depicts the change in a function in terms of the power of a number.

For example, if y = log (x), then the function changes according to the power of 10 of the variable.
As per the definition of a logarithmic function, if n =  logb (x), then x = bn. This leads to an obvious conclusion that the logarithm of a number (or positive variable) to the same base is always 1.

If the base of a logarithmic function is 'e', the exponential constant, then it is called as a natural logarithm and denoted as ln. Scientific calculators give the value of a logarithm of any number with respect to these two bases.

If the base of a logarithmic function is 10, then it is called a common logarithm and denoted just as a log. Logarithmic functions are preferred due to the fact that the operations with logarithms are simple, as the properties of logarithmic functions are interesting and helpful. Hence let us first review that.

Properties of Logarithmic Functions

While the properties of logarithm functions are applicable for any type, for better clarity, we will describe the basic form of a logarithmic function f(x) = logb (x).

The logarithm of product of two functions is the same as the sum of the logarithms of individual functions.
That is, logb (x*y). = logb (x).+ logb (y)

The logarithm of the quotient of two functions is the same as the difference of the logarithms of individual functions.
That is, logb $(\frac{x}{y})$. = logb (x).- logb (y)

The logarithm of a power function is the same as power times the logarithms of the base function.
That is, logb (x)n. = n*logb (x)

These properties help a lot in simplifying complicated functions.

A logarithmic function can be expressed with a different base as per the following property. logb (x).= $[\frac{log_{a}(x)}{log_{a}(b)}]$
The above property helps us to find the logarithm of any number to any base, by changing the base to e or 10 and then using the calculators.

Derivatives of Logarithmic Functions

Let us start by finding the derivative of a logarithmic function from the first principle. Assume the simplest form of a logarithmic function f(x) = ln (x). Even if the function is transformed, the derivative can still be figured out by using differentiation rules.

f(x) = ln (x). Let h be an infinitesimally small change in the variable, that is h is far less than x.

f(x + h) = ln (x + h)

Hence, f(x + h) – f(x) = ln (x + h) – ln (x) = $\frac{[(x+h)]}{(x)}$ = ln $[1+(\frac{h}{x})]$, applying logarithmic rules and simplifying.

As per the definition of logarithmic series,
ln $\frac{[(x+h)]}{(x)}$ = ln $[1+(\frac{h}{x})]$ = $(\frac{h}{x})$ - $(\frac{(h^{2})}{2x^{2}})$ + $(\frac{h^{3}}{3x^{3}})$ - $(\frac{h^{4}}{4x^{4}})$ + ………

Therefore. $\frac{[f(x+h)-f(x)]}{[h]}$ = $(\frac{1}{x})$ - $(\frac{h}{2x^{2}})$ + $(\frac{h^{2}}{3x^{3}})$ - $(\frac{h^{3}}{4x^{4}})$ + ………

Now, as per the definition of derivatives,
$f(x)=limit h\rightarrow 0 of \frac{[f(x+h-f(x)]}{[h]}=limit h\rightarrow 0[(\frac{1}{x})-(\frac{h}{2x^{2}})+(\frac{h^{2}}{3x^{3}})-(\frac{h^{3}}{4x^{4}})+......]=\frac{1}{x}$

Thus, the derivative of ln (x) = $\frac{1}{x}$.

Let us also find the derivative if the base of the logarithm is not the natural base but another constant say, b.

Let g(x) = logb (x)

But by the change of base rule of logarithms, $g(x) = \frac{[in(x)]}{[in(b)]}$.

Therefore, $g(x) = [\frac{(1)}{(x*in b)}]$, since ln (b) is a constant.

Thus, the derivative of logb (x) = $[\frac{(1)}{(x*inb)}]$.

Logarithmic Functions Examples

We have been referring to logarithmic functions in its simplest form f(x) = logb (x). The most general form of a logarithmic function is f(x) = logb [g(x)] + c, where g(x) is any polynomial function.

As said earlier, a function which varies slowly even for a large variation in the variable is reframed more conveniently as a logarithmic function. It is also called as a logarithmic scale

For example, the intensity of sound is expressed in decibels, which is a logarithmic function. Similarly the intensity of earth quakes and hydrogen ion concentration, measure of entropy, certain statistical data etc. are all logarithmic functions.

Solving Logarithmic Functions

Given below are some problems on solving logarithmic functions

Graphing Logarithmic Functions

As explained earlier, a logarithmic function changes far less compared to the change in the variable. Hence the graph of a logarithmic function will be horizontally broader and less steep  vertically. Make a table of values that are compatible to plot the points on a grid. A logarithmic graph will always have a vertical asymptote for the value of the variable which makes the function undefined. Draw a smooth curve joining the points that are plotted and taking the vertical asymptote as guidance.

Consider an example of f(x) = log (x + 10). The function is not defined at x = -10, because (x + 10) becomes 0 there. Therefore, the y = -10 is the vertical asymptote.

When x = 0, y = 1, when x = 30, y = 1.6, when x = 60, y =1.8 and when x = 90, y = 2. So the compatible ordered pairs to plot on the grid are, (0, 1), (30,1.6), (60,1.8) and (90, 2). The graph is drawn as shown below.

Inverse of Logarithmic Functions

Let us first review what an inverse function is. if f(x) and g(x) are two functions which are inverse to each other, then (fog)(x) = (gof)(x) = x.

Let f(x) be a logarithmic function defined as f(x) = logb(x). Consider g(x) is its inverse function. Then, as per the invertible conditions,
(fog)(x) = x

That is, logb[g(x)] = x,

or, g(x) = bx, which is an exponential function with the same base.

In general, the inverse function of logarithmic functions are exponential functions.