The Power Rule: Exponents Become Multipliers
log(a^n) = n times log a
Equation
y = log(x^3) - 1*log(x)
Graph
Table
| x | y |
|---|---|
| 1 | 0 |
| 2 | 1.39 |
| 3 | 2.2 |
| 4 | 2.77 |
| 5 | 3.22 |
What this lesson covers
What you do
You shape the function y = log(x^n) - k*log(x) and watch the graph answer.
Challenges to clear
- log(x³) unfolds into copies of log x: with n = 3, slide k until the difference is 0 at x = 2.
- Still 0 at x = 4 — the exponent simply jumps down as a multiplier.
- A higher power inside the log: slide n to 5.
- Now slide k until the difference is 0 at x = 3. The exponent simply steps down and becomes the multiplier.
Check yourself
Why does the exponent 'n' move outside the logarithm as a multiplier?
If log(2) ≈ 0.301, what is the approximate value of log(2^5)?
- Because log(x^n) represents adding log(x) to itself n times. — correct
- Because exponents and logarithms are inverse operations that cancel out.
- Because the base of the logarithm is always 10.
- Because multiplying by n increases the value of the logarithm.
- 1.505 — correct
- 0.301
- 5.000
- 0.060
Think about it
- log(x³) = log(x·x·x) = log x + log x + log x. How many log x's is that?