Exponents are the numbers or variables which are raised to either constants or variables. It is also a repeated multiplication. |

**Given below are the important rules of exponents and logarithms.**

**Exponential Rules**$x^{n}.x^{m}$ = $x^{n + m}$

$x^{m-n}$ = $\frac{x^{m}}{x^{n}}$, $m$ > $n$

$(x.y)^{n}$ = $x^{n}$ . $y^{n}$

$\frac{x^{m}}{x^{n}}$ = $\frac{1}{x^{n-m}}$

($x^{n})^{m}$ = $x^{nm}$

**Logarithmic Rules**ln e$^{x}$ = $x$

e$^{lnx}$ = $x$, $x$ > 0

log$_{a}$x = $\frac{log_{m}x}{log_{m}a}$

log$_{m}$1 = 0 as m$^{0}$ = 1

log$_{m}$a = p as m$^{p}$ = a

log$_{m}m^{x}$ = x as m$^{x}$ = m$^{x}$

log$_{m}$xy = log$_{m}$x + log$_{m}$y

log$_{m}\frac{x}{y}$ = log$_{m}$x - log$_{m}$y

log$_{m}x^{n}$ = n (log$_{m}$x)

**There are different ways to solve the exponential equations. Some of the exponential equation problems are solved below.**

**Example 1:**Solve 3$^{x}$ = 5

**Solution:**For the given problem we need to find the value of x.

Take log on both sides

log 3$^{x}$ = log 5

$x$ log 3 = log 5

$x$ = $\frac{log\ 5}{log\ 3}$

$x$ = 1.465

**Example 2:**Solve e$^{x}$ = 45

**Solution:**To find the value of x we need to use "ln".

As natural log is the inverse of exponential function.

ln e$^{x}$ = ln 45

$x$ = ln(45)

$x$ = 3.8067

Now e$^{3.807}$ = 45.0151 $\sim$ 45

**Example 3:**Solve 5(3)$^{x+2}$ = 25

**Solution:**(3)$^{x+2}$ = 5

Take log on both sides

log (3)$^{x+2}$ = log 5

($x$ + 2) log 3 = log 5

$x$ + 2 = $\frac{log\ 5}{log\ 3}$

$x$ = 1.465 - 2

$x$ = - 0.535

**Example 4:**Solve 3$^{x+2}$ = 3$^{5}$

**Solution:**For the given problem we see that the bases are same so it is easy to equate the exponents.

$x$ + 2 = 5

$x$ = 5 - 2

$x$ = 3

Therefore, the value of the variable $x$ is 3

**Given below are some of the derivatives of exponential and logarithmic functions.**

$\frac{d}{dx}$ e$^{x}$ = $e^{x}$

$\frac{d}{dx}$(log$_{e}$x) = $\frac{1}{x}$

$\frac{d}{dx}$(e$^{f(x)}$) = $f '(x)$ e$^{f(x)}$

$\frac{d}{dx}$ (ln $f(x)$) = $\frac{f'(x)}{f(x)}$

$\frac{d}{dx}$ log $x$ = $\frac{1}{x}$

e$^{ln(x)}$ = $x$ for $x$ > 0 and ln(e$^{(x)})$ = $x$ for all real $x$

Graphing exponential and logarithmic functions is pretty simple, most of the log graphs tend to have the same shape as a square-root graph. Let us explain with the help of examples:

**Example 1:**Graph the function $f(x)$ = e$^{2x}$

**Solution:**

The function e$^{2x}$ is always positive and is located in quadrants 1 and quadrants 2.

As the value of $x$ increases the value of $f(x)$ also increases. The function e$^{2x}$ is an increasing function.

**Example 2:**Graph the function $f(x)$ = ln (5x)

**Solution:**

The function $f(x)$ = ln (5x) is located in quadrants 1 and 4. Notice that the graph is not touching y axis.

This function is always positive and $x$ can never equal zero. The graph will cross the $x$ axis at 1. This is also an increasing function.

**Applications of exponential and logarithmic functions can be seen in finance, physics, natural sciences etc., Some predefined rules are used to solve them.**

**For example:**Solve 18 = 2 (3

^{y}) ā 2

Move all the terms without $y$ in their exponent to one side of the equation. So, our equation now becomes: 20 = 2 (3

^{y}).

Taking log on both sides we get,

If you take log base 10, 20 = 2 (3

^{y}) becomes log

_{10}(10 $\times$ 2) = log

_{10}(2 (3

^{y})).

Simplifying equation by using:

Log (a

^{b}) = b log (a) and log

_{10}(a $\times$ b) = log

_{10}(a) + log

_{10}(b). We get,

=> Log

_{10}(2) + Log

_{10}(10) = log

_{10}(2) + y (log

_{10}3)

Or $y$ = $\frac{1}{ log\ 3}$