Intermediate Algebra

What you'll learn

• Apply exponent rules (product, quotient, power, negative, and zero exponents)
• Understand logarithms as the inverse of exponents: log_b(x) = y means b^y = x
• Identify and extend arithmetic and geometric sequences
• Find the nth term of arithmetic sequences: a_n = a_1 + (n−1)d
• Find the nth term of geometric sequences: a_n = a_1 × r^(n−1)

Key concepts

Exponent rules: x^a × x^b = x^(a+b), x^a / x^b = x^(a−b), (x^a)^b = x^(ab), x^0 = 1, x^(−n) = 1/x^n. Logarithms reverse exponentiation: log₂(8) = 3 because 2³ = 8. Key log rules: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), log(a^n) = n·log(a). An arithmetic sequence has a constant difference between terms (2, 5, 8, 11 with d = 3). A geometric sequence has a constant ratio (3, 6, 12, 24 with r = 2). These advanced topics typically appear in the last 10-15 questions of the Math section and are worth 2-4 questions per test.

Pro tips

• If you see a logarithm question and freeze, convert it to exponential form: log_b(x) = y → b^y = x.
• For sequences, finding the common difference (arithmetic) or common ratio (geometric) is always the first step.
• Negative exponents mean 'reciprocal,' not 'negative number'. This is a common misconception the ACT exploits.