THE COMPOUND // FIELD NOTES // WEEK 30
Logarithms arrive, with the rest of the inequalities
Field Book
Lessons 112 to 115
Method
Paper first, every step
Review 4 runs this week, the last one. A logarithm tells the exponent: that one sentence is the whole invention.
What you have to be able to do
- Solve quadratic inequalities of the less than kind and nonlinear inequalities
- Evaluate logarithms and antilogarithms base ten
- Solve exponential equations by matching bases
The procedures
Quadratic inequalities, less thanField Book Lessons 110 and 112
- Find the boundary points from the matching equation.
- Less than selects between the boundaries: one segment. Test a point to confirm.
LogarithmsField Book Lesson 113
- log N tells the base ten exponent of N: log 1000 = 3 because 103 = 1000.
- The antilog reverses it: antilog 3 = 1000.
Nonlinear inequalitiesField Book Lesson 114
- y > f(x) selects the region above the curve, y < f(x) below.
- Dash the curve for strict inequalities, then shade.
Exponential equationsField Book Lesson 115
- Write both sides as powers of the same base, then set the exponents equal.
- Compound interest runs on this: the exponent is the time.
Worked: a log by definition
- Solve 10x = 1000.
- 1000 is 103, so x = 3.
- In log language: log 1000 = 3.
Terms on the record
| Term | Meaning |
|---|
| logarithm | the exponent the base needs to produce the number |
| antilogarithm | the number recovered from a given logarithm |
| exponential equation | an equation with the unknown in an exponent |