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In this lesson, we will look at what are logarithms and the relationship between exponents and logarithms.

Logarithms can be considered as the inverse of exponents (or indices).

Definition of Logarithm

If

a^{x}=ysuch thata> 0,a≠ 1 then log_{a}y=x

a^{x}=y↔ log_{a}y=x

Exponential Form

y=a^{x}

Logarithmic Form

log

_{a}y=xRemember: The logarithm is the exponent.

**Example:**

Convert the following exponential form to the logarithmic form:

a) 4^{2} = 16

b) 2^{5} = 32

c)

** Solution: **

a) 4^{2} = 16

2 = log_{4} 16 (*the log is the exponent*)

b) 2^{5} = 32

5 = log_{2} 32

**Example:**

Convert the following logarithmic form to exponential form

a) 3 = log_{2} 8

b) 2 = log_{5} 25

c)

** Solution: **

a) 3 = log_{2} 8

2^{3} = 8

b) 2 = log_{5} 25

5^{2} = 25

Take note of the following:

- Since
*a*^{1}=*a*, log_{a}*a*= 1 - Since
*a*^{0}= 1, log1 = 0_{a} - Log
0 is undefined_{a} - Logarithms of negative numbers are undefined.
- The base of logarithms can be any positive number except 1.
- Logarithms to the base 10 are known as common logarithms and are represented by log
_{10}or log. - Logarithms to the base e are known as natural logarithms and are represented by log
_{e}or ln.

Function Notation with Logs and Exponentials

Function notation is used frequently in science to express functions that contain logs and exponents. We learn to use function notation with logs and exponentials in order to solve problems such as computing compounding interest. We can solve these problems written in function notation with logs and exponentials using techniques from solving exponential and log equations.

This video provides an introduction to solving logarithmic and exponential equations. A methodology is introduced so that students will have some type of basic strategy for solving these types of equations.

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