The Quotient Rule: A Subtraction in Disguise
log(a/b) is just log a minus log b
Equation
y = log(10/x) - (log(10) - 3*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(10/x) - (log(10) - k*log(x)) and watch the graph answer.
Challenges to clear
- log(10/x) is log 10 minus exactly ONE log x: slide k to 1 — difference 0 at x = 2.
- Still 0 at x = 5: a quotient inside the log is a subtraction outside.
- Your own numerator: slide a to 8, so the log holds log(8/x).
- log 8 minus exactly ONE log x — slide k until the difference is 0 at x = 2. A quotient inside is a subtraction outside.
Check yourself
Why does dividing inside a logarithm correspond to subtracting outside?
If log(1000) = 3 and log(10) = 1, what is the value of log(1000/10) using the quotient rule?
- Because dividing powers with the same base subtracts their exponents, and logarithms extract those exponents. — correct
- Because logarithms are linear functions that distribute over division.
- Because the log of a fraction is always negative, requiring subtraction.
- Because division is the inverse of multiplication, which corresponds to addition.
- 2 — correct
- 30
- 1
- 0.33
Think about it
- log(10/x) = log 10 − log x. How many log x's get subtracted?