# Exponentials and Logarithms

## Introduction to Exponentials and Logarithms

- Exponential functions are functions of the form
**y = a**^{x}, a > 0 - The basic property of an exponential function is that a quantity increases or decreases at a rate proportional to its current value.
- The logarithmic function is the inverse of an exponential function. It is denoted by
**y = log**_{a}(x) - The
**base ‘a’**of the logarithmic function is the**base of the exponential function**to which it acts as an inverse. - In logarithmic functions, log(a.b) = log(a) + log(b) and log(a/b) = log(a) - log(b). It is essential to grasp these properties.

## Important Properties of Exponentials

- The function
**y = a**has a^{x}**horizontal asymptote**at y=0. For a > 1, this is an exponential growth, while for 0 < a < 1, it’s an exponential decay. - The function is always positive:
**a**for all values of x.^{x}> 0 - The function
**y = a**has a horizontal asymptote at y = b. This vertical shift is significant.^{x}+ b **a**and^{x+y}= a^{x}.a^{y}**(a**^{x})^{y}= a^{x.y}

## Main Properties of Logarithms

- Inverse relation:
**log**and_{a}(a^{x}) = x**a**. These equations express the inverse nature of logarithmic and exponential functions.^{loga(x)}= x - Change of base rule:
**log**. This rule allows logarithms to change their base to facilitate calculations._{b}(a) = log_{c}(a) / log_{c}(b) - Power rule:
**log**. This property allows the manipulation of exponents within logarithmic expressions._{a}(b^{n}) = n.log_{a}(b) **log**regardless of base a, and_{a}(1) = 0**log**regardless of base a._{a}(a) = 1

## Solving Exponential and Logarithmic Equations

- To solve exponential equations, like
**a**, rewrite the equation in logarithmic form:^{x}= b**x = log**._{a}(b) - To solve logarithmic equations such as
**log**, convert to exponential form: a_{a}(x) = b^{b}= x. - Remember to check the domain of your answers. For example,
**log**doesn’t exist for x ≤ 0._{a}(x)

## Applications of Exponentials and Logarithms

- Exponential and logarithmic functions are used in many scientific fields, like physics, biology, and computer science.
- They model phenomena where things grow or decay at a rate proportional to their size, such as populations (growth) or radioactive materials (decay).
- Logarithmic scales, like the Richter scale, are used for phenomena with huge variations in size.