Quick answer

Template: lognew(x) = logold(x) / logold(new) with one auxiliary base throughout.

Formula

  • Drill daily
  • Verify with b^y
  • Use hero tool for speed

Introduction

Repetition builds speed only when each repetition includes domain checks and a written ratio line.

Treat these drills as logarithm base conversion practice, not new theory.

If you want gentler introductory rows first, start with the change of base formula examples; for calculator accuracy tips between drills, read the log base conversion calculator methods.

The hero tool stays at calculator in the home page hero on every visit.

Practice habits and prompts

Write the ratio before calculator entry so partial credit is possible.

Circle restrictions: x positive; bases positive, not 1.

Keep a one-line verification beside every answer (bk check or tool).

Educational exercises should mix base 2, 10, and e notation.

Statistics transfers sometimes give ln; converting to log10 is a common follow-up step.

Computer science drills often hide base 2 in Big-O discussions while calculators stay base 10.

Formula

  • log_b(x) = log_a(x) / log_a(b)

Every drill below uses the same skeleton.

Round only at the end; handheld keys and browser tools can disagree slightly if you round early.

Practice set

  1. Attempt without calculator when possible. Simplify perfect powers first.
  2. Compute with consistent auxiliary base. Use all log or all ln in the ratio.
  3. Verify. Compare with calculator in the home page hero.
  4. Log mistakes. Note whether errors were domain, base swap, or rounding.

Eight prompts with solutions outlined

1) x = 32, convert log2(32) to base 10: expect 5 / log10(2).

2) x = 1/8, base 2 to base 10: log10(1/8) = -3 / log10(2).

3) x = 27, base 3 to base 9: log9(27) = 3/2.

4) x = 50, use ln to get log10(50).

5) x = 1000, compare log10(1000) with log2(1000) via ratio (they differ).

6) x = 7, estimate log5(7) with calculator ln form.

7) x = 1, any valid base: logb(1) = 0.

8) x = 10, base e to base 2: ln(10)/ln(2).

Log which mistakes were domain errors versus base swaps so the next drill set is targeted.