Logarithm - Proof – Multiplication Law

• Recap of logarithm
• Multiplication law of logarithms
• Proof of multiplication law

Recap of logarithm

• Logarithm is the inverse operation of exponentiation.
• It helps us solve problems involving exponential growth or decay.

Multiplication law of logarithms

• The multiplication law of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
• Mathematically:
  • log(ab) = log(a) + log(b), where a > 0, b > 0

Proof of multiplication law

Step 1: Assume two positive numbers a and b. Step 2: Let x = log(ab) Step 3: By definition of logarithm, we can rewrite x = log(ab) as 10^x = ab. Example:

• Let’s take a = 10 and b = 100.
• Logarithm of a = log(10) = 1
• Logarithm of b = log(100) = 2
• The product of a and b = 10 * 100 = 1000. Using the multiplication law of logarithms, we can write:
• Logarithm of the product = log(ab) = log(10 * 100) = log(1000) Step 4: Rearranging the equation, we get 10^x = 1000. Step 5: Solving for x, we find x = 3. So, log(ab) = 3.

Summary

• The multiplication law of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
• The proof of this law involves assuming two positive numbers and using the definition of logarithm and exponential function.

Slide 11

Logarithm - Proof – Multiplication Law

Recap of logarithm

• Logarithm is the inverse operation of exponentiation.
• It helps us solve problems involving exponential growth or decay.

Multiplication law of logarithms

• The multiplication law of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
  • log(ab) = log(a) + log(b), where a > 0, b > 0

Proof of multiplication law

• Step 1: Assume two positive numbers a and b.
• Step 2: Let x = log(ab).
• Step 3: By definition of logarithm, we can rewrite x = log(ab) as 10^x = ab.

Example

• Let’s take a = 10 and b = 100.
• Logarithm of a = log(10) = 1.
• Logarithm of b = log(100) = 2.
• The product of a and b = 10 * 100 = 1000. Using the multiplication law of logarithms, we can write:
• Logarithm of the product = log(ab) = log(10 * 100) = log(1000).

Step 4

• Rearranging the equation, we get 10^x = 1000.

Step 5

• Solving for x, we find x = 3. So, log(ab) = 3.

Summary

• The multiplication law of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
• The proof of this law involves assuming two positive numbers and using the definition of logarithm and exponential function.

Slide 14

Logarithm - Proof – Division Law

Recap of logarithm

• Logarithm is the inverse operation of exponentiation.
• It helps us solve problems involving exponential growth or decay.

Division law of logarithms

• The division law of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
  • log(a/b) = log(a) - log(b), where a > 0, b > 0

Proof of division law

• The proof is similar to the proof of the multiplication law.

Example

• Let’s take a = 100 and b = 10.
• Logarithm of a = log(100) = 2.
• Logarithm of b = log(10) = 1.
• The quotient of a and b = 100 / 10 = 10. Using the division law of logarithms, we can write:
• Logarithm of the quotient = log(a/b) = log(100 / 10) = log(10).

Step 4

• Rearranging the equation, we get 10^x = 10.

Step 5

• Solving for x, we find x = 1. So, log(a/b) = 1.

Summary

• The division law of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
• The proof of this law is similar to the proof of the multiplication law and relies on the definition of logarithm and exponential function.

Slide 17

Logarithm - Proof – Power Law

Recap of logarithm

• Logarithm is the inverse operation of exponentiation.
• It helps us solve problems involving exponential growth or decay.

Power law of logarithms

• The power law of logarithms states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.
  • log(a^n) = n * log(a), where a > 0

Proof of power law

• The proof of this law involves the use of the multiplication law of logarithms and applying it to a number raised to a power.

Example

• Let’s take a = 10 and n = 3.
• The number raised to the power = 10^3 = 1000. Using the power law of logarithms, we can write:
• Logarithm of the number raised to the power = log(a^n) = log(10^3) = log(1000).

Step 4

• The product of the power and logarithm = n * log(a) = 3 * log(10) = 3 * 1 = 3. So, log(a^n) = 3.

Summary

• The power law of logarithms states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.
• The proof of this law involves using the multiplication law of logarithms and applying it to a number raised to a power.

Slide 20

Logarithm - Proof – Change of Base Formula

Recap of logarithm

• Logarithm is the inverse operation of exponentiation.
• It helps us solve problems involving exponential growth or decay.

Change of base formula

• The change of base formula allows us to find the logarithm of a number in a different base by using logarithms in a known base.
  • log_b(x) = log_d(x) / log_d(b), where x > 0, b > 0, and d is any base.

Proof of change of base formula

• The proof of this formula involves using the definition of logarithm and applying the power rule.

Change of base formula (contd.)

• Let’s consider an example:
  • Find log₄16 using the change of base formula.
  • We can choose any base for the calculation.
  • Let’s choose base 2.
• Using the change of base formula:
  • log₄16 = log₂16 / log₂4 = log₂16 / 2.
  • We know that 2⁴ = 16.
• So, log₂16 = 4.
• Substituting this value into the formula, we get:
  • log₄16 = log₂16 / 2 = 4 / 2 = 2.

Summary

• The change of base formula allows us to find the logarithm of a number in a different base by using logarithms in a known base.
• The proof of this formula involves using the definition of logarithm and applying the power rule.
• It can be useful when we need to calculate logarithms in bases that are not easily accessible, such as base e or base 10.

Slide 23

Logarithm - Natural Logarithm

Definition

• The natural logarithm, denoted as ln(x), is a logarithm with base e, where e is the mathematical constant approximately equal to 2.71828.
• The natural logarithm is widely used in mathematics, especially in calculus and exponential growth/decay problems.

Properties of natural logarithm

• Similar to other logarithms, the natural logarithm has properties like:
  • ln(xy) = ln(x) + ln(y)
  • ln(x/y) = ln(x) - ln(y)
  • ln(x^n) = n * ln(x)

Example

• Let’s take x = 2 and y = 3.
• Using the properties of natural logarithm:
  • ln(xy) = ln(2*3) = ln(6)
  • ln(x) + ln(y) = ln(2) + ln(3)
• Evaluating further, we find that ln(6) is approximately 1.79176, ln(2) is approximately 0.69315, and ln(3) is approximately 1.09861.
• Therefore, ln(xy) = ln(6) ≈ 1.79176 and ln(x) + ln(y) ≈ 0.69315 + 1.09861 ≈ 1.79176.

Summary

• The natural logarithm, denoted as ln(x), is a logarithm with base e.
• It has properties similar to other logarithms, including the multiplication, division, and power laws.
• The natural logarithm is widely used in mathematics, especially in calculus and exponential growth/decay problems.

Slide 26

Logarithm - Applications

Real-world applications

• Logarithms have numerous applications in various fields, such as:
  • Science: Logarithmic scales are used to measure pH, earthquake magnitudes, sound levels, etc.
  • Finance: Logarithmic returns are used to calculate investment performance.
  • Computer science: Logarithms are used in algorithms and data structures.
  • Biology: Logarithms help measure biological growth rates and DNA sequencing.

Example 1: pH Scale

• The pH scale measures the acidity or alkalinity of a solution.
• It is based on the concentration of hydrogen ions (H+) in the solution.
• The pH value is calculated using the logarithmic formula: pH = -log[H+].
• For example, a solution with a hydrogen ion concentration of 10⁻⁵ has a pH value of 5.

Example 2: Compound Interest

• Compound interest is the concept of earning interest on both the initial amount and the accumulated interest.
• The formula for compound interest is: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
• Logarithms are used to determine the time required for an investment to double or to calculate the interest rate required to reach a specific target amount.

Example 3: Exponential Decay

• Logarithms are used to model exponential decay, where the quantity decreases over time.
• The formula for exponential decay is: A = A₀ * e^(-kt), where A is the final amount, A₀ is the initial amount, e is the base of natural logarithms, k is the decay constant, and t is the time.
• Logarithms allow us to solve for the decay constant or the time required for the quantity to decrease to a certain level.

Summary

• Logarithms have various applications in science, finance, computer science, biology, and many other fields.
• Examples of applications include the pH scale, compound interest, and exponential decay.
• Understanding logarithms and their properties is crucial for solving real-world problems and making accurate calculations.
• Logarithms are a powerful tool for analyzing and interpreting data.