All calculatorsLogarithm (Log) Calculator
Compute logarithms in any base: y = logb(x). Use the quick base toggle (10, e, 2, or custom), get steps (change-of-base), quick picks, and a mini log curve visual.
Background
A logarithm answers: “what power do I raise the base to, to get the argument?” If by = x, then y = logb(x). Logs are only defined for x > 0, and the base must satisfy b > 0 and b ≠ 1.
How to use this calculator
- Choose the base using the quick toggle (10, e, 2, or custom).
- Enter the argument x (must be > 0).
- Click Calculate to get the value, steps, and the curve visual.
How this calculator works
- Definition: If by = x, then y = logb(x).
- Change of base: logb(x) = ln(x)/ln(b) (works for any valid base).
- Domain: x > 0, b > 0, and b ≠ 1.
Formula & Equation Used
Log definition: y = logb(x) ⇔ by = x
Change of base: logb(x) = ln(x)/ln(b)
Example Problems & Step-by-Step Solutions
Example 1 — log10(1000)
- Base b = 10, argument x = 1000.
- Because 103 = 1000, we get log10(1000) = 3.
Example 2 — log2(1/8)
- x = 1/8 = 2-3.
- So log2(1/8) = -3.
Example 3 — Change of base
- To compute log3(10), use ln(10)/ln(3).
- This gives a decimal approximation.
Frequently Asked Questions
Q: Why must x be positive?
Because by is positive for any real y when b>0.
Q: Why can’t the base be 1?
Because 1y = 1 for all y, so it can’t uniquely “undo” exponentials.
Q: What’s the difference between log and ln?
log(x) usually means base 10, while ln(x) is base e.
