Exponents & Logarithms

A logarithm asks the reverse of an exponent. log₂8 means how many times you multiply 2 to reach 8, and the answer is 3, so the two are sides of one coin. By turning multiplication into addition, it tames huge numbers and spans wide ranges like loudness and earthquakes. Here you drag the base to watch the curve climb and see a log scale space multiplication evenly.

Exponent Intuition — Repeated Multiplication

Base a2

💡 Exponentiation is repeated multiplication

①2^3 = 2×2×2 = 8: 'multiply 2 three times'

②Larger base → explosive growth

③Base > 1 grows, 0 < base < 1 shrinks

Logarithm Intuition — The Inverse Question

💡 Logs ask 'how many times do I multiply?'

①log_2(8) = 3 → 'how many times multiply 2 to get 8?' → 3 times

②Multiplications appear evenly spaced on a log scale

③Earthquake magnitude, decibels, pH — all log scales

Laws of Exponents & Logs

Exponent Laws

a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m-n

Same base product → add exponents

Definition of Log

a^x = N ⟺ x = log_a N

How many times must a be raised to give N

Log Laws

log_a MN = log_a M + log_a N

Log of a product = sum of logs

Change of Base

log_a b = log_c blog_c a

Any base c works for conversion

Exponent ↔ Log Relationship

🔗 Exponents and logs are mirror images

①Exponent: base + exponent → value (2^3 = 8)

②Log: base + value → exponent (log_2 8 = 3)

③Knowing one immediately gives the other

④Graphs are reflections across y = x

Core Identities

a^{log_a N} = N, log_a a^x = x

Exponent and log are mutual inverses

Wrap-up

Core

log_a N = x ⟺ a^x = N

Logarithm is the inverse of exponentiation

🎯 Exam Points

①Definition: a^x = N ⟺ log_a N = x

②Conditions: a > 0, a ≠ 1, N > 0

③Log laws: product→sum, quotient→difference, power→coefficient

④Change of base: log_a b = log_c b / log_c a

⑤Distinguish common log (log₁₀) from natural log (ln)

Worked Examples & Past Exam

Example 1

Evaluate log₂ 8 + log₂ 4.

A sum of same-base logs combines into the log of a product.

log₂ 8 + log₂ 4 = log₂ (8 × 4) = log₂ 32

Since 32 = 2⁵, the log equals the exponent 5.

log₂ 32 = log₂ 2⁵ = 5

▸ 5

Combine same-base logs into a product, then write it as a power.

Example 2

If log₂ 3 = a, express log₂ 24 in terms of a.

Factor 24 = 2³ × 3.

log₂ 24 = log₂ (2³ × 3) = log₂ 2³ + log₂ 3

log₂ 2³ = 3 and log₂ 3 = a.

= 3 + a

▸ a + 3

Split the argument into a power of the base times the remaining factor.

2023 CSAT Math type, adapted

If log₃ 5 = a and log₃ 4 = b, express log₃ 80 in terms of a and b. (Note: 80 = 16 × 5)

①a + b

②a + 2b

③2a + b

④ab

⑤2a + 2b

▸ ② a + 2b

Factor 80 = 4² × 5.

log₃ 80 = log₃ (4² × 5) = log₃ 4² + log₃ 5

log₃ 4² = 2 log₃ 4 = 2b and log₃ 5 = a.

= 2b + a = a + 2b

Exponential Function

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