Exponential & Logarithmic Functions
“Exponents are the world of multiplying; logs rewind it into the world of adding.”
The formula
y = aˣ ⇔ x = logₐ yHow to read it: a multiplied by itself x times gives y; conversely, how many times you multiply a to reach y is the log, x
- a
- — the base — a positive number other than 1 (the number being multiplied repeatedly)
- aˣ
- — a raised to the power x (the exponential)
- logₐ y
- — how many times you multiply a to get y (the logarithm)
The hook
Exponentials and logarithms are mirror images — one asks 'how many times did you multiply?', the other hands back the answer.
In plain words
aˣ multiplies a by itself x times; logₐ y asks back 'how many times must I multiply a to reach y?'. Exponent and log undo each other.
The intuition
The exponential explodes upward through multiplication; the log rewinds that explosion, growing as slowly as a digit count. Think of it as a translator between the world of multiplying (exponents) and the world of adding (logs).
How it's built
For a>1 the exponential y=aˣ shoots up as x grows; the logarithm is its inverse, that same graph reflected across y=x. That's why logₐ(aˣ)=x and a^(logₐ y)=y cancel out.
Example
Since 2³=8, running it backward gives log₂ 8 = 3. 'Multiply 2 three times to get 8' read as a log is 'the base-2 log of 8 is 3'.
Common mistake
log(x+y) is NOT log x + log y. What the log turns into addition is multiplication — log(xy)=log x + log y. Mix these up and every log calculation collapses.
Where it's used
Decibels of sound, the Richter scale of earthquakes, pH acidity, compound interest, bacterial growth — anything that grows 'by a factor' speaks the language of exponents and logs.
Where it came from
Napier invented logarithms in the 1600s to 'turn multiplication into addition and cut down on calculation'. Before calculators, log tables literally lengthened astronomers' working lives.
Prerequisites
Quick check
What is log₂ 8?
- 1
- 2
- 3✓
- 8
Practice
Find log₂ 16.
Answer: 4
- How many times do you multiply 2 to get 16?
- 2⁴ = 16, so log₂ 16 = 4
Takeaway: A log asks back for the exponent.
Find log₁₀ 1000.
Answer: 3
- How many times do you multiply 10 to get 1000?
- 10³ = 1000, so log₁₀ 1000 = 3
Takeaway: The base-10 log is really a 'digit-count' sense.
Solve the equation 3ˣ = 81.
Answer: 4
- Write 81 as a power of 3: 81 = 3⁴
- Same base, so x = 4
Takeaway: For exponential equations, match the base and compare exponents.
Write one line explaining why logₐ 1 = 0.
Answer: undefined
- It asks 'how many times to multiply a to get 1?'
- a⁰ = 1 (multiply zero times → 1)
- So logₐ 1 = 0
Takeaway: For any base, the log of 1 is always 0.