Surds, Indices & Logarithms
Mental model
Indices are fast forward, logs are rewind.
Indices represent repeated multiplication. Surds are just indices in disguise (fractional powers). Logarithms are the inverse: asking 'what power do I need to raise the base to get this number?'
Formula
Indices: a^m x a^n = a^(m+n). a^m / a^n = a^(m-n). (a^m)^n = a^(mn). a^0 = 1. a^(-n) = 1/a^n. a^(1/n) = nth root of a. Surds: sqrt(a) x sqrt(b) = sqrt(a x b). sqrt(a) / sqrt(b) = sqrt(a/b). sqrt(a) + sqrt(b) != sqrt(a+b). Logs: log_a(x) means a^? = x. log_a(mn) = log_a(m) + log_a(n). log_a(m/n) = log_a(m) - log_a(n). log_a(m^n) = n x log_a(m). log_a(a) = 1. log_a(1) = 0. Base change: log_a(b) = log_c(b) / log_c(a). Flip: log_a(b) x log_b(a) = 1.
For indices, multiplying same-base terms adds exponents. For logs, base change allows converting to base 10 or base e. Surds merge under multiplication and division, but NOT under addition (sqrt(4+9) != sqrt(4)+sqrt(9)).
Worked example
Problem: Simplify: (2^3 x 4^2) / (8^2).
- Convert to same base: 4 = 2^2, so 4^2 = (2^2)^2 = 2^4. 8 = 2^3, so 8^2 = (2^3)^2 = 2^6.
- Expression = (2^3 x 2^4) / 2^6 = 2^(3+4) / 2^6 = 2^7 / 2^6 = 2^(7-6) = 2^1 = 2.
- Sanity-check: 8 x 16 / 64 = 128 / 64 = 2. Matches.
Answer: 2.
Problem: If log_2(32) = x and log_4(64) = y, find x + y.
- log_2(32): 2^? = 32 = 2^5. So x = 5.
- log_4(64): 4^? = 64. 4 = 2^2, 64 = 2^6. So (2^2)^? = 2^6 -> 2? = 6 -> ? = 3. So y = 3.
- x + y = 5 + 3 = 8.
- Sanity-check: log_4(64) via base change = log_2(64) / log_2(4) = 6 / 2 = 3. Correct.
Answer: x + y = 8.
Shortcuts
- Infinite Nested Roots: For sqrt(a + sqrt(a + ...)), if 'a' is a product of two consecutive integers n x (n+1), the answer is (n+1). (sqrt(12 + sqrt(12 + ...)). 12 = 3 x 4. Answer = 4.)
- Logarithmic Base Swap: log_a(b) x log_b(a) = 1. You can freely flip base and argument by taking the reciprocal. (If log_2(3) = x, then log_3(2) = 1/x.)
Common mistakes
- Distributing roots over addition: sqrt(a^2 + b^2) does NOT equal a + b. Roots can only be distributed over multiplication and division.
- Log of a sum: log(a + b) does NOT equal log(a) + log(b). The product rule is log(a x b) = log(a) + log(b).
Glossary
- indices
- Small numbers that tell you how many times to multiply a base number by itself.
- fractional
- Made up of fractions or pieces of a whole, rather than whole numbers.
- exponents
- Another word for indices or powers, indicating repeated multiplication.
Recall questions
- What is x^0, given x != 0?
- Simplify log_a(a).
- Is log(1) defined?
Questions & answers
Find the value of x if log_3(x) + log_9(x) = 3.
9
Approach: Change base: log_9(x) = log_3(x) / log_3(9) = log_3(x) / 2. So log_3(x) + 0.5 x log_3(x) = 3. 1.5 x log_3(x) = 3. log_3(x) = 2. x = 3^2 = 9.
Simplify sqrt(72) - sqrt(18)
3 x sqrt(2)
Approach: sqrt(72) = sqrt(36 x 2) = 6 x sqrt(2). sqrt(18) = sqrt(9 x 2) = 3 x sqrt(2). 6 x sqrt(2) - 3 x sqrt(2) = 3 x sqrt(2).
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