Logarithm Properties and Rules

This page delves deeper into the properties of logarithms, presenting key rules for manipulating logarithmic expressions and solving equations.

Product Rule: The logarithm of a product is the sum of the logarithms of the factors. log_a(b · c) = log_a(b) + log_a(c)

Example: log_2(1024 · 32) = log_2(1024) + log_2(32) = 10 + 5 = 15

Quotient Rule: The logarithm of a quotient is the difference of the logarithms. log_a(b / c) = log_a(b) - log_a(c)

Example: log_5(125 / 25) = log_5(125) - log_5(25) = 3 - 2 = 1

Power Rule: The logarithm of a power is the product of the exponent and the logarithm of the base. log_a(b^n) = n · log_a(b)

Example: log_2(3^2) = 2 · log_2(3)

Change of Base Formula: To change the base of a logarithm, use the formula: log_a(b) = log_c(b) / log_c(a)

Highlight: The zmiana podstawy logarytmu (change of base) formula is crucial for calculating logarithms with non-standard bases using a calculator.

Logarithm of 1: The logarithm of 1 with any base is always 0. log_a(1) = 0

Vocabulary: Kiedy logarytm jest równy 0 (When is a logarithm equal to 0) - This occurs when the argument of the logarithm is 1.

These properties are fundamental for solving complex logarithmic equations and simplifying expressions. Practice with various logarytmy zadania (logarithm exercises) to master these concepts.