Topic 4: Logarithm

What is Logarithm?

A logarithm is the power to which a number must be raised in order to get some other number.

For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:

log 100 = 2

because,

10² = 100

Properties for Condensing Logarithms

Property 1: 0=log_a  1= Zero-Exponent Rule

Property 2: 1 = log_a a

Property 3: log_a x + log_a y log_a (xy) – Product Rule

Property 4: log_a x – log_a y = log_a (x/y) – Quantient Rule

Property 5: y log_a x = log_a x³ – Power Rule

What is Exponent in Logarithm? 

An exponent refers to the number of times a number is multiplied by itself.

For example, 2 to the 3rd (written like this: 2³) means:

2 x 2 x 2 = 8.

2³ is not the same as 2 x 3 = 6.

*Remember that a number raised to the power of 1 is itself.

For example,

a¹ = a

5¹ = 5.

There are some special cases:

(1)  a⁰ = 1

When an exponent is zero, as in 6⁰, the expression is always equal to 1.

a⁰ = 1

6⁰ = 1

14,356⁰ = 1

(2)  a^-m = 1 / a^m

When an exponent is a negative number, the result is always a fraction.

Fractions consist of a numerator over a denominator. In this instance, the numerator is always 1. 

To find the denominator, pretend that the negative exponent is positive, and raise the number to that power, like this:

a^-m = 1 / a^m

6^-3 = 1 / 6³

You can have a variable to a given power, such as a³, which would mean a x a x a. 

You can also have a number to a variable power, such as 2^m, which would mean 2 multiplied by itself times. 

We will deal with that in a little while.

Example of Logarithm

Example 1: solve log₃ (9x+2) = 4

Log₃ (9x+2) = 4

9x + 2 = 3⁴

9x + 2 =81

X= 79/9

Final answer: log₃ (9x+2) = 4 is x=79/9

Example 2: solve log₄ (2x+1) = log₄ (x+2) – log₄ 3

log₄ (2x+1) = log₄ (x+2) – log₄ 3

log₄ (2x+1) = log₄ (x+2/3)

2x + 1 = x+2/3

3 (2x+1) = x +2

6x + 3 = x+2

x= -1/5

Final answer= no solution.

Example 3: solve log(5x-11) = 2

log(5x-11) = 2

5x – 2 = 10²

5x- 2 = 100

X=102/5

Final answer: log(5x-11) = 2 is x 102/5



SEE THIS VIDEO! (learn something from this video).....

SOLVE THIS QUESTIONS?

[Q1] Log₂ (8) = 3

[Q2] Log₅ = (625) = 4

[Q3] Log₄ (16) = 2

[Q4] Log₈ (x) = 2

[Q5] Log₈ (64) = x

[Q6] Log (6.11)

[Q7] Log (3.2³)

[Q8] Log (6/11)⁵

[Q9] Log 2⁴/5

[Q10] Log (6/5)