Estimating Logarithms

Mental Math · Axiom Academy

Master mental techniques for approximating logarithms without a calculator The key to estimating logarithms mentally is understanding that log₁₀(x) tells us "what power of 10 gives us x?" Start by identifying which powers of 10 bracket your number. For example, to estimate log₁₀(350): We know 10² = 100 and 10³ = 1000 Since 100 < 350 < 1000, we know 2 < log₁₀(350) < 3 350 is closer to the middle-upper range, so log₁₀(350) ≈ 2.5-2.6 2. Mental Interpolation Technique Once you've bracketed your logarithm, you can refine your estimate using linear interpolation on a log scale. The trick is knowing a few key reference points. Let's estimate log₁₀(350) more precisely: log₁₀(350) = log₁₀(3.5) + log₁₀(10²) = log₁₀(3.5) + 2 Since 3.5 is between 3 and 5, we interpolate: log₁₀(3.5) ≈ 0.544 Here are practical shortcuts you can use in your head for fast approximations:

This is the written version of the interactive lesson above. See the full Mental Math course.