Quick answer

Pattern: write logb(x) = loga(x) / loga(b), substitute, simplify.

Formula

  • Base 2 ↔ base 10
  • Base 5 ↔ base 10
  • ln form for science classes

Introduction

Examples turn the abstract ratio into something you can grade, debate, and verify. Each sample below states variables before any calculator keystrokes.

Try each example by hand first, then compare with the hero tool so you notice when simplification is possible.

When you finish this set, the logarithm base conversion examples add mixed-base drills for exam review, and the change of base formula with natural log explains when ln(x)/ln(b) is the fastest scientific route.

Use calculator in the home page hero to confirm decimals.

How to read each example

Setup names x, the original base a, and the target base b in words before symbols.

Work shows the division line with the same auxiliary base on both sides.

Result states logb(x) in simplified form when algebra allows.

Educational exercises should include at least one perfect-power case (integer log) and one decimal case.

Scientific calculator examples should note whether log means base 10 in your course.

Check domain before starting: x positive; bases valid.

Reference

  • log_b(x) = log_a(x) / log_a(b)

Every example below uses this skeleton; only numbers and bases change.

Science classes that print ln first still use the same ratio with a = e in both numerator and denominator.

Worked conversions

  1. Copy the template. Write logb(x) = loga(x) / loga(b) before substituting numbers.
  2. Substitute and simplify. Reduce fractions when logs collapse to integers.
  3. Verify. Use bk checks or calculator in the home page hero.

Six worked samples

Base 2 to base 10: x = 8, log10(8) = log2(8) / log2(10) = 3 / log2(10).

Base 5 to base 10: x = 625, log10(625) = 4 / log10(5) because log5(625) = 4.

Natural log: x = 20, log10(20) = ln(20) / ln(10) when you choose e as auxiliary base.

Scientific calculator: log3(81) = log(81) / log(3) = 4 / log(3) = 4 exactly.

CS style: log2(1024) = 10; converting to base 10 gives 10 / log10(2).

Caution: x = 0.01 with base 10 is valid (negative log) but still requires x > 0.

Repeat one ln-based row by hand if your next unit leans on natural logarithms.