Learn: Logarithms & Exponential Growth

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The idea

A logarithm is the undo button for powers. $\log_b(a)$ asks: “ $b$ to what power gives $a$ ?” Growth says “start, then multiply by 2 every year” — the log says “how many doublings did that take?” The two undo each other, exactly like + and −.

Worked example

Every step explained — why we do it, and what happens.

1
$\log_2{8} = \,?$
Why: Read the log question out loud: “2 to what power gives 8?” A log is the inverse of a power — the answer is an exponent.
What happens: We are hunting the exponent, not the number.
2
$2^{?} = 8$
Why: Rewrite the log as a power question — that is the definition: $\log_2 8 = ?$ means $2^? = 8$ .
What happens: Now it is a question we already know how to read.
3
$2 \times 2 \times 2 = 8$
Why: Count the factors: three twos multiply to 8.
What happens: The exponent is hiding in the count.
4
$\log_2{8} = 3$
Three twos, so the exponent is 3.
5
$2^3 = 8$
Why: The check: run the power forward and confirm the log undid it correctly.
What happens: 2³ = 8 ✓ — the log and the power cancel each other, like +4 and −4.

Real-world example

A bacteria culture starts with 5 cells and doubles every 3 hours. How many cells after 9 hours? And how many hours until it reaches 320 cells? (The second question needs the log as the undo button.)

1
$N = 5 \times 2^{t/3}$
Why: Translate the story: start with 5, and every 3 hours the count multiplies by 2. t/3 counts the doublings.
What happens: A growth formula: input t, output the number of cells.
2
$N = 5 \times 2^{9/3} = 5 \times 2^3$
Why: Plug in t = 9: nine hours is exactly 3 doubling periods.
What happens: Only the exponent needs computing now.
3
$= 5 \times 8 = 40$
2³ = 8, then 5 × 8 = 40 cells.
4
$5 \times 2^{t/3} = 320$
Why: Now work backwards: which t makes the count 320? This is the log-shaped question.
What happens: The growth formula, flipped around.
5
$2^{t/3} = 64 \Rightarrow \frac{t}{3} = \log_2 64 = 6$
Why: Apply the undo button: “2 to what power gives 64?” Six twos multiply to 64, so t/3 = 6.
What happens: The log turned a growth question into plain arithmetic.
6
$t = 18$
Why: Multiply by 3: 6 × 3 = 18 hours.
What happens: Check: $5 \times 2^6 = 5 \times 64 = 320$ ✓ — 18 hours is the answer.

Common glitch

Two classic glitches:

1.Reading $\log_2 8$ as a fraction

Wrong: $\log_2 8 = 2 \div 8$ — that is a division, not a log

Right:Ask “2 to what power gives 8?” — the answer is the exponent 3.

2.Adding the growth instead of multiplying

Wrong:Start 5, double twice → “5 + 5 + 5 = 15”

Right:Each doubling multiplies: $5 \times 2 \times 2 = 20$ .

General Calculator — Evaluate powers and logarithms with the scientific keys.

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Common glitch

Two classic glitches:

1.Reading $\log_2 8$ as a fraction

Wrong: $\log_2 8 = 2 \div 8$ — that is a division, not a log

Right:Ask “2 to what power gives 8?” — the answer is the exponent 3.

2.Adding the growth instead of multiplying

Wrong:Start 5, double twice → “5 + 5 + 5 = 15”

Right:Each doubling multiplies: $5 \times 2 \times 2 = 20$ .

Learn: Logarithms & Exponential Growth

The idea

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Logarithms & Exponential Growth

The idea

Worked example

Real-world example

Common glitch

Try a related calculator