How to Calculate Logarithms Without a Calculator
Before calculators, engineers worked out logarithms with a few memorised values and some simple rules. The same tricks still work and make logs far less mysterious. This guide shows how to calculate log without calculator help, accurate to three or four decimal places.
What a logarithm is
log10(x) answers: “10 to what power gives x?” So log 1,000 = 3, because 10³ = 1,000. Natural log, ln, uses the base e ≈ 2.71828 instead of 10.
Values worth memorising
| x | log10 x |
|---|---|
| 2 | 0.3010 |
| 3 | 0.4771 |
| 7 | 0.8451 |
| 10 | 1 |
With these four, the log rules below give you almost any number.
The log rules
- Product: log(ab) = log a + log b
- Quotient: log(a ÷ b) = log a − log b
- Power: log(an) = n × log a
How to calculate log without calculator: worked examples
log 6
6 = 2 × 3, so log 6 = 0.3010 + 0.4771 = 0.7781. The true value is 0.7782; the last digit differs only because of rounding.
log 5
5 = 10 ÷ 2, so log 5 = 1 − 0.3010 = 0.6990.
log 72
72 = 8 × 9 = 2³ × 3², so log 72 = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572. The true value is 1.8573, so four memorised values already get you within one ten-thousandth.
Characteristic and mantissa
For large or small numbers, write them in scientific notation. The power of ten is the characteristic, and the log of the rest is the mantissa:
log 4,500 = log(4.5 × 10³) = log 4.5 + 3
4.5 = 9 ÷ 2, so log 4.5 = 0.9542 − 0.3010 = 0.6532, and log 4,500 = 3.6532.
Natural log without a calculator
Convert from base 10 with one multiplication:
ln x = 2.3026 × log10 x
ln 2 = 2.3026 × 0.3010 = 0.6931. For numbers close to 1, the series ln(1 + x) ≈ x − x²/2 + x³/3 works well: ln 1.1 ≈ 0.1 − 0.005 + 0.00033 = 0.09533, against a true value of 0.09531.
Change of base
To find a log in any base, divide two base-10 logs:
logb x = log x ÷ log b
log2 10 = 1 ÷ 0.3010 = 3.3219. Notice that log2 10 and log10 2 are reciprocals of each other.
To check your hand-worked answers, the log calculator gives the logarithm in any base along with the natural and common logs.
Common mistakes
- log(a + b) is not log a + log b. The product rule only works for multiplication.
- Dropping the characteristic. log 4,500 is 3.6532, not 0.6532.
- Logs of zero or negatives. They are undefined for real numbers.
Reciprocals like the change-of-base pair are explained in how to find the multiplicative inverse.
For another formula you can work out by hand, see how to calculate the distance formula.
When you need speed over practice, an online calculator gives the same logs to more decimal places.