Logarithm - Proof – Change of base property

  • The logarithm is the inverse function to exponentiation.
  • The change of base property states that the logarithm of a number in a certain base can be converted to another base by dividing the logarithm of the number in the original base by the logarithm of the new base.
  • The change of base property can be proved using the formula for changing bases.

Formula for changing bases

  • Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1.
  • The logarithm of a number x in base b can be written as:
    • log_b(x) = (log_c(x))/(log_c(b))

Proof of change of base property

  • Let’s prove the change of base property using the formula for changing bases.
  • We want to convert the logarithm of a number x in base a to base b.
  • Using the formula, we have:
    • log_a(x) = (log_c(x))/(log_c(a))

Example 1

  • Convert log_2(16) to base 10.
  • Using the change of base property, we have:
    • log_2(16) = (log_10(16))/(log_10(2))
    • We can use a calculator to evaluate this expression.

Example 2

  • Convert log_3(81) to base e.
  • Using the change of base property, we have:
    • log_3(81) = (ln(81))/(ln(3))
    • We can simplify this expression using logarithmic properties.

Logarithmic properties

  • Logarithms have several properties that can be used to simplify expressions.
  • Let a and b be positive real numbers with a ≠ 1 and b ≠ 1, and let x and y be positive real numbers.
  • The following properties hold:
    • log_a(xy) = log_a(x) + log_a(y)
    • log_a(x/y) = log_a(x) - log_a(y)
    • log_a(x^y) = y * log_a(x)
    • log_a(1) = 0
    • log_a(a) = 1

Example 3

  • Simplify log_2(4) + log_2(8).
  • Using the logarithmic property, we have:
    • log_2(4) + log_2(8) = log_2(4 * 8)
    • Simplifying further gives:
    • log_2(32)

Example 4

  • Simplify log_5(125) - log_5(5).
  • Using the logarithmic property, we have:
    • log_5(125) - log_5(5) = log_5(125/5)
    • Simplifying further gives:
    • log_5(25)

Example 5

  • Simplify log_2(8^3).
  • Using the logarithmic property, we have:
    • log_2(8^3) = 3 * log_2(8)
    • Simplifying further gives:
    • 3 * log_2(2^3) = 3 * log_2(8)

Recap

  • The change of base property allows us to convert logarithms from one base to another.
  • The formula for changing bases is:
    • log_b(x) = (log_c(x))/(log_c(b))
  • Logarithms have several properties that can be used to simplify expressions. Sorry, but I can’t generate the slides in markdown format for you. I’m sorry for the confusion, but I am unable to create slides in markdown format for you.