Logarithm & Exponent Calculator

Worked Example: log₁₀(1000)

1. Calculate log base 10 of 1000.
2. Ask: 10 to what power equals 1000?
3. 10¹ = 10, 10² = 100, 10³ = 1000.
4. Therefore log₁₀(1000) = 3.

Final Answer: log₁₀(1000) = 3

Formula Explained

A logarithm answers: "to what power must the base be raised to get this number?"

logb(x) = y means b^y = x

Common logs use base 10 (log), natural logs use base e ≈ 2.718 (ln). Key rules: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), log(aⁿ) = n·log(a).

Example: What is log₂(32)?

1. Calculate log₂(32) — "2 raised to what power gives 32?"
2. Try powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32
3. 2⁵ = 32, so log₂(32) = 5
4. Verify: 2⁵ = 32 ✓

Answer: log₂(32) = 5

Example: What is 3⁴?

1. 3⁴ = 3 × 3 × 3 × 3
2. 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81
3. Answer: 81

Formula Explained

• Definition: log_b(x) = y means b^y = x
• log (no base shown) = log base 10
• ln = natural log, base e ≈ 2.71828
• Change of base: log_b(x) = ln(x) / ln(b)
• Key rule: log and exponent are inverse operations

Frequently Asked Questions

What is a logarithm in simple terms?

A logarithm answers: 'to what power must the base be raised to get this number?' log₂(8) = 3 because 2³ = 8.

What is the difference between log and ln?

log (common log) uses base 10. ln (natural log) uses base e ≈ 2.718. Both are used in science and mathematics.

What is Euler's number (e)?

e ≈ 2.71828 is a mathematical constant. It is the base of natural logarithms and appears in growth and decay formulas.

How do I use the change of base formula?

log_b(x) = log(x) / log(b). This lets you calculate any base using your calculator's log or ln button.

What is an antilogarithm?

The inverse of a logarithm. If log₁₀(x) = 3, then the antilog is 10³ = 1000. It reverses the logarithm operation.

Why is log(0) undefined?

Because no power of any positive base can equal zero. As x approaches 0 from the right, log(x) approaches negative infinity.

Related Calculators