log_b(xy) = log_b(x) + log_b(y)
This law states that the logarithm of a product is equal to the sum of the logarithms of its factors.
Example 1:
log₂(8) = log₂(2 × 4) = log₂(2) + log₂(4)
log_b(x/y) = log_b(x) – log_b(y)
This law states that the logarithm of a quotient is equal to the difference of the logarithms of its numerator and denominator.
Example 2:
log₃(27) = log₃(81 / 3) = log₃(81) – log₃(3)
log_b(x^y) = y * log_b(x)
This law states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base.
Example 3:
log₅(125) = log₅(5³) = 3 * log₅(5)
Additional Example:
Simplify log₂(32) – log₂(8):
Using the Law of Logarithm Division:
log₂(32) – log₂(8) = log₂(32 / 8) = log₂(4) = 2
It’s important to note that these laws hold for any base of the logarithm (b), not just the examples provided.
Logarithms are commonly used in various fields, such as science, engineering, and computer science. They help solve problems related to exponential growth, time complexity analysis, and sound intensity, among others.