Simplifying mathematical expressions
Logarithms
Main equations
| loga(bc) | = | loga(b) + loga(c) | |
| loga(b / c) | = | loga(b) − loga(c) | |
| loga(bc) | = | loga(b) × c | |
| loga(b1/c) | = | loga(b) / c |
Special cases – one variable
| loga(a) | = | 1 | |
| loga(1) | = | 0 | |
| loga(1 / a) | = | −1 |
Special cases – two variables
| loga(ab) | = | 1 + loga(b) | |
| loga(a / b) | = | 1 − loga(b) | |
| loga(b / a) | = | loga(b) − 1 | |
| loga(1 / b) | = | −loga(b) | |
| loga(ab) | = | b | |
| loga(a1/b) | = | 1 / b | |
| loga(√b) | = | loga(b) / 2 | |
| loga(∛b) | = | loga(b) / 3 | |
| logab(a) + logab(b) | = | 1 | |
| 1 − logab(a) | = | logab(b) |
Example: 1 − log10(5) = log10(2)
Special cases – three variables or more
| loga(abc) | = | b + loga(c) | |
| loga(ab / c) | = | b − loga(c) | |
| loga(b) / loga(c) | = | logc(b) | |
| logab(ac) + logab(bd) | = | 1 + logab(c) + logab(d) |
Examples:
- log10(81) / log10(3) = log3(81) = 4
- log10(14) + log10(15) = 1 + log10(7) + log10(3)
Involving the golden ratio
The golden ratio (phi): φ = (1 + √5) / 2 ≈ 1.618
Moving square roots and phi from the denominator to the numerator.
No variables
| 1 | / | φ | = | φ − 1 | |
| 1 | / | (φ + 1) | = | 2 − φ | |
| 1 | / | (φ − 1) | = | φ |
One variable
| 1 | / | (aφ) | = | (φ − 1) | / | a | |
| 1 | / | (aφ + 1) | = | (a(φ − 1) − 1) | / | (a(a − 1) − 1) | |
| 1 | / | (aφ − 1) | = | (a(φ − 1) + 1) | / | (a(a + 1) − 1) | |
| 1 | / | (φ + a) | = | (a − (φ − 1)) | / | (a(a + 1) − 1) |
Examples:
- 1 / (3φ + 1) = (3φ − 4) / 5
- 1 / (φ + 3) = (−φ + 4) / 11
Two variables
| 1 | / | (aφ + b) | = | (a(φ − 1) − b) | / | (a2 − ab − b2) |
Example: 1 / (3φ + 5) = (8 − 3φ) / 31
Other expressions
Moving square roots from the denominator to the numerator.
One variable
| 1 | / | √a | = | √a | / | a | |
| 1 | / | (a + √a) | = | (a − √a) | / | (a(a − 1)) | |
| 1 | / | (a − √a) | = | (a + √a) | / | (a(a − 1)) | |
| 1 | / | √(a + √a) | = | √(a(a − 1)(a − √a)) | / | (a(a − 1)) | |
| 1 | / | √(a − √a) | = | √(a(a − 1)(a + √a)) | / | (a(a − 1)) | |
| (√(2a) + √2) / 2 | = | √((a + 1) / 2 + √a) | |||||
| (√(2a) − √2) / 2 | = | √((a + 1) / 2 − √a) | |||||
Examples:
- 1 / √(3 + √3) = √(18 − 6√3) / 6
- (√10 + √2) / 2 = √(3 + √5)
Two variables
| 1 | / | (a + √b) | = | (a − √b) | / | (a2 − b) | |
| 1 | / | (a − √b) | = | (a + √b) | / | (a2 − b) | |
| 1 | / | √(a + √b) | = | √((a2 − b)(a − √b)) | / | (a2 − b) | |
| 1 | / | √(a − √b) | = | √((a2 − b)(a + √b)) | / | (a2 − b) | |
| 1 | / | (√a + √b) | = | (√a − √b) | / | (a − b) | |
| 1 | / | (√a − √b) | = | (√a + √b) | / | (a − b) | |
| (√(ab) + √a) / √(2a) | = | √((b + 1) / 2 + √b) | |||||
| (√(ab) − √a) / √(2a) | = | √((b + 1) / 2 − √b) | |||||
| √(a + √b) + √(a − √b) | = | √(2a + 2√(a2 − b)) | |||||
| √(a + √b) − √(a − √b) | = | √(2a − 2√(a2 − b)) | |||||
Examples:
- 1 / √(3 + √7) = √(6 − 2√7) / 2
- (√21 + √3) / √6 = √(4 + √7)
- √(3 + √5) − √(3 − √5) = √2
Three variables
| 1 | / | (a + √(b + √c)) | = | (a − √(b + √c)) × (a2 − b + √c) | / | ((a2 − b)2 − c) | |
| 1 | / | (a + √(b − √c)) | = | (a − √(b − √c)) × (a2 − b − √c) | / | ((a2 − b)2 − c) | |
| 1 | / | (a − √(b − √c)) | = | (a + √(b − √c)) × (a2 − b − √c) | / | ((a2 − b)2 − c) | |
| 1 | / | (a − √(b + √c)) | = | (a + √(b + √c)) × (a2 − b + √c) | / | ((a2 − b)2 − c) | |
| 1 | / | √(a + √(b + √c)) | = | √(a + √(b + √c)) × (a − √(b + √c)) × (a2 − b + √c) | / | ((a2 − b)2 − c) | |
| 1 | / | √(a + √(b − √c)) | = | √(a + √(b − √c)) × (a − √(b − √c)) × (a2 − b − √c) | / | ((a2 − b)2 − c) | |
| 1 | / | √(a − √(b − √c)) | = | √(a − √(b − √c)) × (a + √(b − √c)) × (a2 − b − √c) | / | ((a2 − b)2 − c) | |
| 1 | / | √(a − √(b + √c)) | = | √(a − √(b + √c)) × (a + √(b + √c)) × (a2 − b + √c) | / | ((a2 − b)2 − c) |
Examples:
- 1 / (2 + √(5 + √7)) = (√(5 + √7) − 2) × (√7 − 1) / 6
- 1 / √(2 + √(5 + √7)) = √(2 + √(5 + √7)) × (√(5 + √7) − 2) × (√7 − 1) / 6