Quick answer

Common log: log(x) = log10(x). Change of base: converts between bases with a ratio.

Formula

  • log key ≠ base-free log
  • Change of base generalizes bases
  • Use ratio when subscripts change

Introduction

Students treat the log key as a universal logarithm button. In most school contexts, it means base 10 unless a problem states otherwise.

Change of base is not a competitor to common log; it is the tool that relates common log to base 2, e, or custom bases.

When homework uses ln heavily, read the natural log change of base form; when you need vocabulary for what change of base means before comparing, start with the definition of the change of base formula.

Test both ideas with calculator in the home page hero using bases 10 and e.

Key differences and calculator use

Common logarithm evaluates log10(x) directly when log means base 10.

Change of base rewrites logb(x) using logs you can already compute.

When to use each: use common log when the problem is already base 10; use change of base when subscripts or context change.

Calculator applications: press log for base 10, ln for base e, and the ratio when you need logb with another b.

Common misconceptions include assuming log(100) equals log2(100), or treating ln as a scaled version of log without a clear ratio.

Misconceptions also appear in word problems that hide base 10 (pH, decibels) while exams suddenly introduce base 2.

Linking the ideas

  • log_b(x) = log(x) / log(b)
  • log(x) = log_10(x) on many calculators

Treat log(x)/log(b) as change of base with auxiliary base 10.

Lab exercises that export ln often still convert with ln(x)/ln(b) when the target base is not e.

Comparison guide

  1. Identify whether the problem is base 10. Look for context words and subscripts.
  2. Apply change of base when subscripts differ. Write the ratio before touching calculator keys.
  3. Avoid mixing keys in one line. Pick log or ln for both numerator and denominator.
  4. Clarify definitions with reading. Write whether log means base 10 in your course before you compare two logarithms.

Misconception checks

log(1000) = 3 in base 10. log2(1000) is not 3; you need change of base to compare them fairly.

ln(100) and log(100) are different numbers; connect them with log10(100)/log10(e) thinking or the natural log ratio.

After fixing misconceptions, practice one ln ratio and one base-10 ratio on the same argument x.