Logarithms are powerful tools for simplifying complex mathematical expressions. They have unique properties that allow us to break down products, quotients, and powers into simpler forms, making calculations easier.

Understanding logarithms is crucial for solving equations and working with exponential functions. We'll explore key properties like the product, quotient, and power rules, as well as the change-of-base formula for practical applications.

Properties of Logarithms

Properties of logarithms

• Product property: $\log_b(M \cdot N) = \log_b(M) + \log_b(N)$ states that the logarithm of a product is equal to the sum of the logarithms of its factors ($8x$ and $x$)
  • Allows for the expansion of logarithmic expressions involving products into sums of logarithms
  • Useful for simplifying and solving equations involving logarithms
• Quotient property: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$ states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator ($27$) and denominator ($9$)
  • Allows for the expansion of logarithmic expressions involving quotients into differences of logarithms
  • Useful for simplifying and solving equations involving logarithms
• Power property: $\log_b(M^n) = n \cdot \log_b(M)$ states that the logarithm of a number raised to a power is equal to the power (exponent) multiplied by the logarithm of the number ($5x$)
  • Allows for the simplification of logarithmic expressions involving powers
  • Useful for solving equations involving logarithms and exponents

Change-of-base formula

• Change-of-base formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ allows for the evaluation of logarithms with any base using a calculator that only has logarithms with base 10 or base $e$ (natural logarithm)
  • Useful when dealing with logarithms of bases other than 10 or $e$
  • Helps in solving equations and simplifying expressions involving logarithms with different bases

Simplification of complex logarithms

1. Identify the properties that can be applied to the given logarithmic expression ($\log_2(x^3)$, $\log_2(y)$, $\log_2(z^2)$)
2. Apply the power property to simplify logarithms with exponents ($3 \cdot \log_2(x)$, $2 \cdot \log_2(z)$)
3. Combine like terms, if possible, to further simplify the expression

• Use the change-of-base formula in combination with other properties to simplify expressions involving logarithms with different bases ($\log_3(x^2)$, $\log_3(y)$, $\log_9(z)$)
  1. Apply the change-of-base formula to convert logarithms with base 9 to base 3 ($\frac{1}{2} \cdot \log_3(z)$)
  2. Apply the power property to simplify logarithms with exponents ($2 \cdot \log_3(x)$)
  3. Combine like terms, if possible, to further simplify the expression

Logarithms and Functions

• Logarithms are inverse functions of exponential functions
• The base of a logarithm determines its specific inverse relationship to an exponential function
• The domain of a logarithmic function is all positive real numbers, while its range includes all real numbers