Indices, Logarithms and Surds

This module covers the fundamental algebraic principles of indices, logarithms, and surds. These concepts are essential for solving complex equations and are a staple of the JAMB Mathematics syllabus.

1. Indices

Indices (or exponents) represent the power to which a base is raised.

• Multiplication Law: a^m * a^n = a^(m+n)
• Division Law: a^m / a^n = a^(m-n)
• Power Law: (a^m)^n = a^(mn)
• Zero Law: a^0 = 1 (where a is not 0)
• Negative Index: a^(-n) = 1 / a^n
• Fractional Index: a^(1/n) = n-th root of a

Standard Form: Expressing numbers in the form A * 10^n, where 1 <= A < 10.

2. Logarithms

Logarithms are the inverse of indices. If y = a^x, then x = log_a(y).

• Product Rule: log_a(XY) = log_a(X) + log_a(Y)
• Quotient Rule: log_a(X/Y) = log_a(X) - log_a(Y)
• Power Rule: log_a(X^n) = n * log_a(X)
• Change of Base: log_a(B) = log_c(B) / log_c(a)

3. Surds

Surds are irrational numbers expressed as roots. E.g., √2, √3.

• Addition/Subtraction: Only like surds can be added (e.g., 2√3 + 4√3 = 6√3).
• Multiplication: √a * √b = √(ab).
• Rationalization: The process of removing a surd from the denominator by multiplying the numerator and denominator by the conjugate. The conjugate of (√a + √b) is (√a - √b).