Slide 1

  • Topic: Logarithm - Examples – Basic problems on logarithms

Slide 2

  • Definition: A logarithm is the inverse function to exponentiation.
  • A logarithm answers the question: “What exponent do I need to raise this base to in order to get this number?”

Slide 3

  • Example 1: Find the value of x in log(base 2) x = 8.
  • Solution: 2 raised to what power gives 8? This gives us x = 3.

Slide 4

  • Properties of Logarithms:
    • log base a of 1 = 0
    • log base a of a = 1
    • log base a of a^x = x
    • log base a of a^x = x
    • log base a of m * n = log base a of m + log base a of n
    • log base a of m/n = log base a of m - log base a of n

Slide 5

  • Example 2: Simplify the expression log(base 3) 27 - log(base 3) 9.
  • Solution: log(base 3) 27 - log(base 3) 9 = log(base 3) (27/9) = log(base 3) 3 = 1.

Slide 6

  • Change of Base Formula: log(base a) b = log(base c) b / log(base c) a

Slide 7

  • Example 3: Find the value of log(base 5) 27 using the change of base formula.
  • Solution: log(base 5) 27 = log(base 10) 27 / log(base 10) 5.

Slide 8

  • Example 4: Solve the equation 2^(x+1) = 8.
  • Solution: 2^(x+1) = 8
    • Rewriting 8 as 2^3, we get: 2^(x+1) = 2^3
    • Therefore, x + 1 = 3
    • Solving for x, we find x = 2.

Slide 9

  • Example 5: Solve the equation log(base 4) (x-3) + log(base 4) x = 2.
  • Solution: log(base 4) (x-3) + log(base 4) x = 2
    • Combining the logs using the product rule, we get: log(base 4) [(x-3) * x] = 2
    • Simplifying, we have: log(base 4) (x^2 - 3x) = 2
    • Exponentiating both sides using base 4, we find x^2 - 3x = 4^2 = 16
    • Rearranging the equation, we get: x^2 - 3x - 16 = 0
    • Factoring or applying the quadratic formula, we find x = -4 or x = 4.

Slide 10

  • Summary:
    • Logarithms are the inverse of exponentiation.
    • Basic problems on logarithms involve finding values, simplifying expressions, and solving equations.
    • The properties of logarithms can be useful in simplifying expressions.
    • The change of base formula allows us to convert logarithms to different bases.

Slide 11

  • Logarithm Rules:
    • log base a of (mn) = log base a of m + log base a of n
    • log base a of (m/n) = log base a of m - log base a of n
    • log base a of (m^p) = p * log base a of m
  • Example 6: Simplify the expression log(base 2) (16/4).
  • Solution: Using the logarithm rules, log(base 2) (16/4) = log(base 2) 16 - log(base 2) 4 = 4 - 2 = 2.

Slide 12

  • Graphs of Logarithmic Functions:
    • The graph of y = log(base a) x is a reflection of the graph of y = a^x across the line y = x.
  • Example 7: Sketch the graph of y = log(base 2) x.
  • Solution: The points (1, 0), (2, 1), (4, 2), etc. can be plotted to create the graph. The graph will approach but never touch the x-axis.

Slide 13

  • Natural Logarithm:
    • The natural logarithm is a logarithm with base e, where e is approximately 2.71828.
  • Example 8: Evaluate ln(e^3).
  • Solution: ln(e^3) = 3 ln e = 3 * 1 = 3.

Slide 14

  • Logarithmic Exponential form:
    • If log(base a) y = x, then a^x = y.
  • Example 9: Rewrite the equation log(base 2) 8 = 3 in exponential form.
  • Solution: log(base 2) 8 = 3 can be rewritten as 2^3 = 8.

Slide 15

  • Solving Logarithmic Equations:
    • Steps:
      1. Rewrite the equation in exponential form.
      2. Solve for the variable.
      3. Check your solution.
  • Example 10: Solve the equation log(base 5) x = 2.
  • Solution: Rewrite the equation in exponential form: 5^2 = x. Therefore, x = 25.

Slide 16

  • Example 11: Solve the equation log(base 3) (x + 2) = 4.
  • Solution: Rewrite the equation in exponential form: 3^4 = x + 2. Simplifying, we find x + 2 = 81. Subtracting 2 from both sides, we get x = 79.

Slide 17

  • Example 12: Solve the equation ln(x + 1) = 2.
  • Solution: Rewrite the equation in exponential form: e^2 = x + 1. Subtracting 1 from both sides, we find x = e^2 - 1.

Slide 18

  • Common Logarithm:
    • The common logarithm is a logarithm with base 10.
  • Example 13: Evaluate log(100).
  • Solution: Since the base is not specified, we assume it to be 10. Therefore, log(100) = 2.

Slide 19

  • Logarithmic Applications:
    • Logarithms have applications in various fields such as biology, finance, computer science, and more.
  • Example 14: In a biology experiment, the number of bacteria grows exponentially with time. The number n of bacteria after t hours can be modeled by the equation n = 10^(0.02t). Find the number of bacteria after 5 hours.
  • Solution: Substitute t = 5 into the equation: n = 10^(0.02 * 5) = 10^(0.1) ≈ 1.259. Therefore, there are approximately 1.259 bacteria after 5 hours.

Slide 20

  • Summary:
    • Logarithmic rules help simplify expressions involving logarithms.
    • Graphs of logarithmic functions are reflections of exponential functions.
    • Natural logarithm has base e.
    • Logarithmic equations can be solved by rewriting in exponential form.
    • Logarithms have applications in various fields.

Slide 21

  • Example 15: Solve the equation log(base 6) (x+1) = log(base 6) (2x-3).
  • Solution: If the logarithms on both sides have the same base, then the arguments must be equal. Therefore, we have x+1 = 2x-3. Simplifying, we find x = 4.

Slide 22

  • Example 16: Solve the equation 5^(2x+1) = 125.
  • Solution: Rewrite 125 as 5^3. Therefore, we have 5^(2x+1) = 5^3. By equating the exponents, we find 2x+1 = 3. Solving for x, we have x = 1.

Slide 23

  • Compound Interest formula: A = P(1 + r/n)^(nt)
  • Example 17: Calculate the amount A received after 5 years on an initial investment of $1000 at an annual interest rate of 5% compounded annually.
  • Solution: Using the compound interest formula, we have A = 1000(1 + 0.05/1)^(1*5) = 1000(1.05)^5 ≈ 1276.28.

Slide 24

  • Common logarithm properties:
    • log(xy) = log(x) + log(y)
    • log(x/y) = log(x) - log(y)
    • log(x^p) = p * log(x)
  • Example 18: Simplify the expression log(1000) - 2 log(10) + log(25).
  • Solution: Using the common logarithm properties, we have log(1000) - 2 log(10) + log(25) = log(10^3) - 2 log(10) + log(5^2) = 3 - 2 + 2 = 3.

Slide 25

  • Example 19: Solve the equation log(x) + log(x + 3) = 2.
  • Solution: Using the logarithm properties, we can combine the logarithms: log(x(x + 3)) = 2. Rewriting in exponential form, we have x(x + 3) = 10^2 = 100. Expanding and rearranging the equation, we find x^2 + 3x - 100 = 0. Solving for x, we get x = -10 or x = 7.

Slide 26

  • Example 20: Solve the equation ln(2x - 4) - ln(5) = ln(3x).
  • Solution: Using the properties of natural logarithms, we can combine the logarithms: ln((2x - 4)/5) = ln(3x). This implies (2x - 4)/5 = 3x. Solving for x, we find x = -2.

Slide 27

  • Change of Base Formula (alternate form): log(base a) b = ln(b) / ln(a)
  • Example 21: Evaluate log(base 8) 64 using the change of base formula.
  • Solution: We can use the change of base formula with base e as the alternate base. log(base 8) 64 = ln(64) / ln(8) ≈ 2.5.

Slide 28

  • Example 22: Determine the logarithmic value x in the equation 3^x = 81.
  • Solution: We can rewrite 81 as 3^4. Therefore, 3^x = 3^4. By equating the exponents, we get x = 4.

Slide 29

  • Example 23: Simplify the expression log(base 3) 27/81.
  • Solution: We can rewrite 27/81 as a power of 3: log(base 3) 3^(-1). Using the logarithm property, we find -1 log(base 3) 3. Since log(base 3) 3 equals 1, the expression simplifies to -1.

Slide 30

  • Summary:
    • Logarithms can be used to solve equations, simplify expressions, and calculate compound interest.
    • Different properties of logarithms can be applied to manipulate logarithmic expressions.
    • The change of base formula allows for conversion between different bases.
    • Application of logarithms extends to compound interest and other mathematical problems.
    • Understanding logarithms is essential for solving various mathematical problems.