How do you report sig figs after logs and exps?

Report significant figures after logarithms and exponentials using propagated uncertainty: for logarithms, the number of decimal places equals the significant figures in the original value; for natural exponentials, match significant digits to the propagated uncertainty in the result.

Think of significant figures like how sure you are about a number, and propagated uncertainty tells you how much that surety changes when you use math functions. When you take a logarithm, you keep as many digits after the decimal point as the original number had significant figures because the log compresses scale but preserves relative precision. For exponentials you first propagate the uncertainty through the function, and then round the final answer so that its last printed digit is about the same size as the uncertainty; that means your reported digits reflect how precise the value actually is after the transformation. If you are unsure, calculate the propagated uncertainty explicitly using derivatives so you can see how many digits remain meaningful.

Students often forget to propagate uncertainty and just apply the same number of significant figures mechanically, which can either overstate or understate precision; always compute the uncertainty through the function to know what to keep. Another common mistake is treating decimal places after a log as significant figures of the logged result instead of matching them to the original argument, so remember the mapping between original significant figures and log decimal places. Rounding too early is a frequent error, so keep extra guard digits during calculation and round only the final result to the appropriate number of significant digits.

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Sig Fig Rules for Logs and Exponentials

1. Treat sig figs as measurement certainty
2. For logs, match decimal places
3. For exponentials, propagate uncertainty first
4. Round so last digit ≈ uncertainty
5. Use derivatives to calculate propagation
6. Keep extra guard digits during calculation

Common Mistakes with Logs and Exponentials

1. Not propagating uncertainty through functions
2. Misreading log decimal places significance
3. Rounding too early

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