Understanding Logarithmic and Exponential Functions

Suppose ‘a’ is any positive real number. In this case, the function f defined by f(x) = ax will be known as the general exponential function. Df(domain of f) = R.

This exponential function f(x) = ax (a>0, x R) has the following properties:

(i)      a0 = 1

(ii)    ax. ay = ax+y for all x, y R

(iii)  (ax)y = axy for all x, y R

(iv)  a-x =  for all x R

The general form of an Exponential function is y = ax

Here,

•  'a' is the base
• 'x' is the exponent

This function takes up an exponent and then calculates its corresponding value.

Let us understand the properties of exponential functions that are important for CBSE board students:

• Exponential functions are both continuous and differentiable throughout their domain
• The natural exponential function  f(x) = ex is unique because its derivative is $\frac{d}{dx} e^x = e^x$
• If a>1, then that exponential function is increasing which indicates an exponential growth. For example, $f\left(x\right) = 2^x$

• If 0 $f\left(x\right) = {\mathrm{(½)}}^x$

• The inverse of  f(x) = ax  is a logarithmic function $f^{-1} \left(x\right) = {\log }_a \left(x\right)$

• For natural exponential function f(x) = ex ,its inverse is natural logarithm:
• The product of two exponentials is ax. ay= ax+y
• The quotient of two exponentials is $\frac{a^x}{a^y} = a^{x-y}$

• The power of an exponential is (ax)y = axy
• The exponential of a product is ax.bx = (ab)x

 

Questions based on the graph of exponential function will be asked in the NEET exam or JEE Main exam.

Let us take a look at some of graphs of exponential functions:

$f\left(x\right) = 12 \cdot { \left( \frac{1}{2} \right) }^x$

 

A logarithmic function is the inverse of an exponential function. If a>0, a ≠ 1 and x>0, then the logarithmic function will be written as $f \left( x \right) = {log}_a \left( x \right)$

Here:

• a>0 and a ≠ 1 (base)
• x>0 (argument)

Let us take a look at the properties of logarithmic functions:

• The domain of a logarithmic function is all positive real numbers (x>0) while its range includes all real numbers (y∈R).
• Graph of a logarithmic function always passes through point (1,0) because ${log}_a \left( 1 \right) = 0$ for valid base a.

• Logarithmic functions  are inverse of exponential functions which means $f \left( x \right) = {log}_a \left( x \right)$  is inverse of  $f\left(x\right) = {\mathrm{a}}^x$

• Product rule states that ${\log }_a (mn) = {\log }_a\left(m\right) + {\log }_a\left(n\right) .$

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