How do sig figs work in logs and exponents?

For logarithms, count significant figures of the original value to set the number of decimal places in the mantissa, and for exponentiation use scientific notation so significant figures apply to the coefficient while exponents are exact.

Think of a logarithm like splitting a number into two parts: a whole number part and a decimal part called the mantissa. The whole number tells you the order of magnitude and does not depend on sig figs, but the decimal part must show as many digits as there are significant figures in the original number, because those decimals carry the precision. For example, log of three point two zero has a mantissa with three digits after the decimal if three point two zero has three significant figures. When you raise a number to a power, write it in scientific notation so the base coefficient carries the significant figures and the exponent is an exact count; multiply or divide coefficients and then round the result to the correct number of significant figures before adjusting the exponent.

A common mistake is treating the whole number part of a log like it follows significant figure rules, which it does not, and the fix is to only control decimal places according to the input precision. Another frequent error is ignoring scientific notation for powers and so accidentally changing the precision of the exponent instead of the coefficient, so always convert to scientific notation and apply sig-fig rules to the coefficient only. A third slip is rounding too early during calculations, so avoid rounding until the final step to keep accuracy.

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Significant Figures in Logs and Exponents

1. Log split into characteristic and mantissa
2. Characteristic not set by sig figs
3. Mantissa digits equal input sig figs
4. Use scientific notation for powers
5. Coefficient carries significant figures
6. Don’t round until final calculation

Common Sig Fig Mistakes in Logs and Exponents

1. Treating characteristic as significant figures
2. Ignoring scientific notation for powers
3. Rounding too early during calculations

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