What is a logarithm?

A logarithm is the inverse operation of exponentiation. If exponentiation answers “What is b raised to the power y?”, logarithms answer the reverse question: “What power y gives me the number x when I raise b to it?”

• Log form: log_b(x) = y
• Equivalent exponential form: b^y = x

These two statements are exactly the same relationship written in two different “languages.” That’s why logs are so useful: they convert multiplication into addition, turn exponent problems into simpler linear relationships, and compress huge ranges of numbers (which is why scientific scales like pH, decibels, and Richter-like scales feel natural).

• Base 10: log(x) or log₁₀(x) (often called the “common log”).
• Base e: ln(x) (the natural log), where e ≈ 2.71828.
• Base 2: log₂(x) (common in computer science and information theory).

• Product rule: log_b(MN) = log_b(M) + log_b(N)
• Quotient rule: log_b(M/N) = log_b(M) - log_b(N)
• Power rule: log_b(M^k) = k · log_b(M)
• Inverse rules: log_b(b) = 1 and log_b(1) = 0

The product/quotient rules are why logarithms show up in scientific notation and “orders of magnitude” reasoning. Multiplying large numbers is hard; adding is easy. Logs let you “pull exponents down” and do arithmetic on them.

Most calculators have built-in ln and log10. To compute log_b(x) for an arbitrary base b, you use:

• Change of base formula: log_b(x) = ln(x) / ln(b)
• or equivalently log_b(x) = log10(x) / log10(b)

This page uses that same idea internally (with careful validation) and then formats the steps so you can see the reasoning. That’s also why base b cannot be 1: ln(1) = 0, and dividing by 0 breaks the formula.

Logarithm examples you can copy

Compute log₁₀(1000). Ask: “10 to what power equals 1000?” Because 10³ = 1000, the answer is 3.

Compute ln(e²). Logs undo exponentials: ln(e²) = 2. In general, ln(e^y) = y.

Compute log₂(32). Since 2⁵ = 32, we get 5. This shows up in questions about doubling: “How many doublings to reach 32?” Answer: five doublings.

Compute log₃(50). This is not a “nice” integer. Use change-of-base: log₃(50) = ln(50) / ln(3). The calculator will display an approximate decimal value and the steps that got you there.

Suppose log₁₀(x) = 4. Convert back to exponent form: 10⁴ = x, so x = 10000. This is exactly what the “Solve for x in b^y = x” mode does.

Suppose you know b³ = 64. Solve for b: take the cube root: b = 64^1/3 = 4. In general, if b^y = x, then b = x^1/y (when y ≠ 0 and x > 0).

What this calculator does behind the scenes

The calculator follows the same logic you would write on paper — it just does it faster and formats it cleanly. Here’s the exact flow:

• For log_b(x), it requires x > 0, b > 0, and b ≠ 1.
• For solving bases, it also checks y ≠ 0 (because b⁰ = 1 for any valid base).
• If inputs fail, it shows a human-friendly error instead of a mysterious “NaN”.

• Compute y: y = ln(x) / ln(b)
• Solve x: x = b^y
• Solve b: b = x^1/y

After calculation, the “Steps” box prints the formula you used, then substitutes the actual numbers, and finally shows the computed result (rounded for readability, while retaining more precision internally). That makes it perfect for quick studying — you see the structure, not just the answer.

People love “I just solved this in 2 seconds” screenshots. The share buttons generate a short text summary (including your mode and result), plus a link back to this page — which is the simplest, cleanest kind of virality.

• If b > 1, then log_b(x) grows slowly as x grows (it “compresses” x).
• If 0 < b < 1, the function decreases as x increases (counterintuitive but totally normal).
• Logs are excellent for comparing ratios: log(a) - log(b) = log(a/b).

Frequently Asked Questions

• What’s the difference between log and ln?

  ln(x) means log base e. Some textbooks use log(x) to mean base 10, while others use it to mean base e. This calculator avoids confusion by letting you set the base explicitly and giving one-click buttons for base 10 and base e.

• Why can’t the base be 1?

  Because 1^y = 1 for every y. There’s no way to uniquely solve “what exponent gives x?” unless x is 1 — it becomes ambiguous. Also, change-of-base uses ln(b), and ln(1) = 0, which would cause division by zero.

• Why does x have to be positive?

  For real-number logarithms, the input must be positive. Negative and zero inputs require complex numbers (advanced math beyond most algebra courses). If you need complex logs, you’ll want a dedicated CAS tool.

• Can logs be negative?

  Yes. If 0 < x < 1 and b > 1, then log_b(x) is negative. Example: log₁₀(0.01) = -2 because 10^-2 = 0.01.

• How many decimals should I use?

  For homework, match your class instructions. For most practical use, 4–8 decimal places is plenty. This calculator prints a readable value and keeps more precision internally for consistent steps.

• Where do logarithms show up in real life?

  Everywhere that scales multiplicatively: sound (decibels), acidity (pH), earthquakes, growth rates, and many “how many times bigger?” comparisons. In finance, logs appear in continuous growth and log-returns; in science, they simplify exponential relationships.

Before you submit your answer

• Did you rewrite the log in exponential form correctly?
• Did you confirm the domain rules (x > 0, b > 0, b ≠ 1)?
• If you used change-of-base, did you write ln(x)/ln(b) (or log10(x)/log10(b))?
• Did you round to the number of decimals requested?

If you want a faster “combo” workflow, pair this with the Exponents Calculator: log solves for the exponent; exponent solves for the value. That’s the same relationship from both sides.