How the change of base formula works
The change of base formula lets you evaluate a logarithm in any base b using only the logarithm functions built into a calculator or programming language — almost always base 10 (common log, often written log) and base e (natural log, ln). It states log_b(x) = log_c(x) / log_c(b), where c is any valid base you choose (c > 0, c ≠ 1). No matter which valid c you pick, the ratio produces the same value of log_b(x).
Where the formula comes from
Let y = log_b(x), which by definition means by = x. Taking log_c of both sides gives log_c(by) = log_c(x), and the power rule of logarithms turns the left side into y · log_c(b) = log_c(x). Dividing both sides by log_c(b) isolates y: y = log_c(x) / log_c(b) — the change of base formula.
Worked example
To find log₂(8) using base-10 logs: log₂(8) = log₁₀(8) / log₁₀(2) ≈ 0.90309 / 0.30103 = 3. Using natural logs instead gives the same answer: ln(8) / ln(2) ≈ 2.07944 / 0.69315 = 3. Either calculation base works because 2³ = 8, and the formula does not care which valid base you route the calculation through.
Using the result correctly
- x must be greater than 0; logarithms of zero or negative numbers are undefined for real results.
- The base b must be greater than 0 and cannot equal 1, since log base 1 is undefined (it would require dividing by log_c(1), which is 0).
- The result can be negative (when 0 < x < 1 and b > 1) or exactly zero (when x = 1, since log_b(1) = 0 for any valid base).
Frequently Asked Questions
What is the change of base formula?
Why do calculators need this formula?
Does the choice of calculation base change the answer?
What input values are not allowed?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |