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 = log_{b} (x), then x = b^{n}. 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.

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 = log

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.

The logarithm of product of two functions is the same as the sum of the logarithms of individual functions.

That is, log

The logarithm of the quotient of two functions is the same as the difference of the logarithms of individual functions.

That is, log

The logarithm of a power function is the same as power times the logarithms of the base function.

That is, log

These properties help a lot in simplifying complicated functions.

A logarithmic function can be expressed with a different base as per the following property. log

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.

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) = log

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 log

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.

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.

Let f(x) be a logarithmic function defined as f(x) = log

(fog)(x) = x

That is, log

or, g(x) = b

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