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.
Drag the base and the curve climbs, and a log scale spaces multiplication evenly.
Exponent Intuition — Repeated Multiplication
When you know the base and the power but not the exponent, the missing number needs a name. That name is log_a N: it is the x that solves a^x = N. Because a product of the same base turns into a sum of exponents, a log turns a long chain of multiplications into addition. The base must satisfy a > 0 and a ≠ 1, and the argument must satisfy N > 0. The slider only allows bases from 1.2 to 4, so the sketch shows growth, not decay. The case 0 < a < 1 is written in the text; it is outside the drawn range.
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Exponential growth curves by base
The upper sketch shows a gold curve y = a^x, with x ticks from 0 to 5. A dashed line sits at height 1, and the title prints the chosen base to one decimal place in y = a^x. When you change base a from 1.2 to 4, the vertical scale is reset to height a^5, so a larger base stays low on the left and then climbs steeply on the right. Below, the same five values a^0, a^1, a^2, a^3, a^4 are marked on two bars. The upper bar is a log scale, so the gaps are even; the lower bar is linear, so larger numbers bunch toward one end. The curve has no extra marker point. Each new base redraws the curve and the ticks in place.
2
💡 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
④This sketch uses a ≥ 1.2. Decay for 0 < a < 1 is the sentence in
③
Logarithm Intuition — The Inverse Question
If you swap the two places in log_a N, you are asking a different question. The first number is the base and the second is the argument; flip them and you have inverted the power. A base of 1, a negative base, or an argument that is 0 or less is not defined, so a familiar-looking symbol is not enough to compute. The rule that turns a log of a product into a sum of logs holds only when the argument is a product. Using that rule on a sum breaks the definition. Dropping the base and reading only the argument also makes the same letters stand for another value.
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Log scale vs linear scale
💡 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
am × an = am+n, am ÷ an = am-n
Same base product → add exponents
Definition of Log
ax = N ⟺ x = loga N
How many times must a be raised to give N
Log Laws
loga MN = loga M + loga N
Log of a product = sum of logs
Change of Base
loga b = logc blogc 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
⑤Common vs natural logs are distinguished later in this chapter