## Continuity and Differentiability

# Exponential and Logarithmic Functions

- The exponential function with positive base b > 1 is the function y = f(x) = b
^{x} - Some of the salient features of the exponential functions.

(i) Domain of the exponential function is**R**, the set of all real numbers.

(ii) Range of the exponential function is the set of all positive real numbers.

(iii) The point (0, 1) is always on the graph of the exponential function (this is a restatement of the fact that b^{0}= 1 for any real b >1).

(iv) Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.

(v) For very large negative values of x, the exponential function is very close to 0. In other words, in the second quadrant, the graph approaches x-axis (but never meets it). - Let b > 1 be a real number. Then we say logarithm of
*a*to base b is*x*if b^{x}= a. - Logarithm of
*a*to base*b*is denoted by log_{b}a. Thus log_{b}*a = x*if b^{x}= a. - Some of the important observations about the logarithm function to any base b > 1 are listed below:

(i) We cannot make a meaningful definition of logarithm of non-positive numbers and hence the domain of log function is**R**^{+}.

(ii) The range of log function is the set of all real numbers.

(iii) The point (1, 0) is always on the graph of the log function.

(iv) The log function is ever increasing, i.e., as we move from left to right the graph rises above.

(v) For*x*very near to zero, the value of log*x*can be made lesser than any given real number. In other words in the fourth quadrant the graph approaches*y*-axis (but never meets it).

### Part1: View the Topic in this video From 20:20 To 46:56

### Part2: View the Topic in this video From 00:40 To 32:42

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**Derivative of exponential functions**:

• \frac{d}{dx}(e^{x})=e^{x}

**Derivative of logarithmic functions**:

• \frac{d}{dx}(\log x)=\frac{1}{x}

• \frac{d}{dx}(a^{x})=a^{x} \log a

• \frac{d}{dx}\left(\log_{a}{x}\right)=\frac{1}{x\cdot \log_{e}{a}}