Mathematics from zero
Logarithms
An exponent answers “multiply 2 by itself 3 times — what do you get?” Answer: 8. A logarithm asks the very same fact from the other end: “how many times must I multiply 2 to reach 8?” Answer: 3. Same relationship, reversed.
After this lesson you can say what a logarithm is, see why it reverses an exponent, compute a simple logarithm, and explain why logarithms grow so slowly.
A logarithm is the reverse of an exponent. You know 2³ = 8: multiply three 2s
and get 8. A logarithm undoes that — it starts from 8 and recovers the exponent,
3. Just as the square root reverses squaring, the logarithm reverses raising to a
power.
A logarithm asks: what exponent reaches this number? We write it log₂(8), read
“log, base 2, of 8”. The small 2 is the base — the number being multiplied. The
answer is the exponent that gets you there: log₂(8) = 3, because 2³ = 8. The base
of the logarithm must match the base of the exponent.
A logarithm grows very slowly — the exact opposite of exponential growth. The curve
above is log₂. As the input runs all the way from 1 to 64, the output only climbs
from 0 to 6. Exponential growth multiplied to explode; the logarithm divides that
explosion back down. Each doubling of the input adds just 1 to the logarithm.
That slowness makes logarithms a measuring tool for huge numbers. Because
log₂ adds 1 every time its input doubles, the logarithm of a number is roughly “how
many doublings it took to get here”. It turns a runaway exponential quantity into a
small, calm number you can compare and reason about — which is why logarithms appear
wherever growth explodes.
Find log₂(16).
The logarithm asks: how many times must you multiply 2 to reach 16?
Count the multiplications. One 2 is 2. Times 2 again is 4. Again is 8. Again is
16. That took four 2s multiplied together: 2 × 2 × 2 × 2 = 16, which is 2⁴.
So the exponent is 4, and log₂(16) = 4. Check by running it forward: 2⁴ = 16. The
logarithm and the exponent are the same fact read in opposite directions.
Why this works
Why does the base matter so much? Because “how many times you multiply” only has a
clear answer once you say multiply by what. log₂(8) is 3, but log₂ of a number
and a log with a different base ask different questions. The base names the step size
of the multiplication; change the base and you change the question, so the base is
always written right there with the log.
Common mistake
A common mistake is reading log₂(8) as 8 divided by 2, or as 2 × 8. It is neither.
A logarithm is not an ordinary operation on the two numbers — it is the exponent
that connects them. log₂(8) asks “2 to what power is 8?” and the answer is 3, not 4
and not 16.
Find log₂(8): how many 2s multiply to 8? Type the value.
Find log₂(2). Type the value.
Find log₁₀(100): how many 10s multiply to 100? Type the value.
Find log₂(16). Type the value.
Find log₁₀(1000). Type the value.
What does log₂(8) = 3 mean?
A logarithm is the reverse of an exponent: it asks how many times you must multiply
the base to reach a given number, and the answer is that exponent. It is written
log₂(8) = 3, because 2³ = 8; the base of the log must match the base of the power.
Logarithms grow very slowly — adding only 1 each time the input doubles — which makes
them a calm scale for measuring explosive growth.