State the law of logarithms. Differentiate between rational numbers and irrational numbers.

The law of logarithms states that for any positive numbers a, b, and c, and any positive real number x:

1. log_a(bc) = log_a(b) + log_a(c)
2. log_a(b/c) = log_a(b) - log_a(c)
3. log_a(b^c) = c * log_a(b)

Rational numbers are numbers that can be expressed as a ratio of two integers, such as 1/2 or -3/4. Irrational numbers, on the other hand, cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal representations. Examples of irrational numbers include √2, π, and e.

Law of Logarithms:

If a, b, and c are positive real numbers such that a ≠ 1 and b ≠ 1, then:

- Product Rule: logₐ(bc) = logₐb + logₐc
- Quotient Rule: logₐ(b/c) = logₐb - logₐc
- Power Rule: logₐ(b^c) = c logₐb
- Change of Base Formula: logₐb = (logₑb) / (logₑa)

Difference between Rational Numbers and Irrational Numbers:

- Rational Numbers: Numbers that can be expressed as a fraction of two integers, p/q, where p and q are integers and q ≠ 0. Examples: 1/2, -3/4, 0.75
- Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers. They are non-terminating and non-repeating. Examples: √2, π, e