Topic- Concept and Application of Logarithm (Class- 9, Experience- 3)

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• MCQ Test
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Exponent

An exponent, also known as a power, indicates how many times a number (the base) is multiplied by itself. It is written as a small number (the exponent) to the upper right of the base number. The general form is:

bⁿ

Here, b is the base and n is the exponent.

Example: 5² = 5×5 = 25 , Here, 5 is the base and 2 is the exponent. This means 5 is multiplied by itself 2 times.

Exponential equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. The general form of an exponential equation is bⁿ = a, where b > 0 and b ≠ 1.

Example:
5^x = 25
Here, b = 5, x = 2, and a = 25

Formulas of exponents

Multiplication of exponents: 

x^m×xn = x^m+n 

Example: 2³×2² = 2³⁺² = 2⁵ = 32


Division of exponents: 

  x^mxⁿ = x^m−n 

Example: 2⁵2³ = 2^5-3 = 2² = 4


Exponentiation of a product: 

(xy)ⁿ = xⁿyⁿ 

Example: (2×3)² = 2²×3² = 4×9 = 36


Exponentiation of a quotient: 

( x  y)ⁿ = xⁿyⁿ 

Example: (23)² = 2²3² = 49


Power of a power: 

(x^m)ⁿ = x^m×n 

Example: (2³)² = 2^3×2 = 2⁶ = 64


Zero exponent: 

x⁰ = 1 (where x ≠ 0) 

Example: 5⁰ = 1


Negative exponent: 

x^-n =  1xⁿ 

Example: 2^-3 = 12³ = 18


Reciprocal exponentiation: 

( x  y)ⁿ = ( y  x)^-n 

Example: ( 2  3)² = ( 3  2)^-2


Logarithm

Logarithm is a mathematical operation that determines the exponent to which a fixed base must be raised to produce a given number. In simple terms, a logarithm of a number x for a base b is the exponent y that satisfies by = xb^y = xby = x. The general form is:

log_b(x) = y

Here,

b is the base and b＞0 and b ≠ 1
x is the argument of the logarithm and x＞0
y is the logarithmic value indicating how many times b is multiplied to get x.

Examples:

log₂(8) = 3 This means 2³ = 8, i.e., the logarithm of 8 to the base 2 is 3.



Types of Logarithm

Common Logarithm: Base 10 is used. log₁₀(x) or log(x) 

Example: 

log₁₀(x) because 10³ = 1000

Natural Logarithm: Base e (approximately 2.71828) is used. log_e(x) or ln(x) 

Example: 

ln(e²) = 2 because e² = e²

Some Formulas Related to Logarithm

Logarithm of 1: 

log_b1 = 0 

because b⁰ = 1


Logarithm of base b: 

log_bb = 1 

because b¹ = b


Product Logarithm Rule: 

log_b(AB) = log_bA + log_bB 

Example: log₂(8×4) = log₂8 + log₂4


Quotient Logarithm Rule: 

log_b(AB) = log_bA − log_bB 

Example: log2(84) = log₂8 − log₂4


Power Logarithm Rule: 

log_b(A^x) = xlog_bA 

Example: log₂(8³) = 3log₂8


Logarithmic Chain Rule: 

log_ab × log_bc = log_ac 

Example: log₂8×log₈64 = log₂64


Inverse Property: 

b^log_ba = a 

Example: 2^log₂8 = 8


Power Exponentiation: 

x^log_by = y^log_bx 

Example: 2^log₂8 = 8^log₈2


Reciprocal Logarithm: 

log_ab =    1log_ba 

Example: log₂8 =    1log₈2


Change of Base Formula: 

log_ax = log_bxlog_ba 

Example: log₂8 = log₁₀8log₁₀2



Using Calculators to Find Logarithm Values:

Using a calculator to determine logarithmic values is convenient and easy. Here are the steps to find different types of logarithms using a calculator:

Finding Common Logarithms (Base 10): log₁₀

1. Turn on the calculator:

Use a scientific calculator. Most calculators have a log⁡\loglog button for base 10 logarithms.

2. Enter the number:

Enter the number whose logarithm you want to find. For example, to find log₁₀1000, enter 1000 in the calculator.

3. Press the log button:

Press the log button. The calculator will display the logarithm of 1000 to base 10.

Example: log₁₀1000 = 3 because 10³ = 1000

Finding Natural Logarithms (Base eee): log_e or ln

1. Turn on the calculator:

Use a scientific calculator. The 'ln' button is for natural logarithms.

2. Enter the number:

Enter the number whose natural logarithm you want to find. For example, to find ln⁡20, enter 20 in the calculator.

3. Press the ln button:

Press the ln button. The calculator will display the natural logarithm of 20.

Example: ln20

Finding Logarithms of Other Bases:

Some scientific calculators allow finding logarithms for different bases.

1. Turn on the calculator:

Use a scientific calculator. Some calculators have a log_b button where b is the base.

2. Enter the number:

Enter the number whose logarithm you want to find. For example, to find log₂16, enter 16 in the calculator.

3. Enter the base:

Enter the base b. For example, if the base is 2, enter 2.

4. Press the log_b button:

Press the log_b button. The calculator will display the logarithm of 16 to base 2.

Example: log₂16 = 4 because 2⁴ = 16