How do I calculate log base b of a number?

Pick the any-base mode, type the true number as x and the base as b. The tool computes it with the change-of-base formula log_b(x) = ln(x) ÷ ln(b). For example log base 2 of 8 is ln(8) ÷ ln(2) = 2.0794 ÷ 0.6931 = 3, because 2 to the power 3 is 8. The breakdown line shows each division so you can check the work, not just trust a single number.

What is the change-of-base formula and why use it?

Most calculators and programming languages only expose ln (base e) and log₁₀ (base 10), not an arbitrary base. The change-of-base formula log_b(x) = ln(x) ÷ ln(b) rewrites any base in terms of one you already have. It works with log₁₀ too: log_b(x) = log₁₀(x) ÷ log₁₀(b). For instance log base 3 of 81 is ln(81) ÷ ln(3) = 4, since 3 to the power 4 is 81.

What is ln and how is it different from log?

ln is the natural logarithm, whose base is the constant e ≈ 2.71828. Plain log usually means the common logarithm with base 10, written log₁₀. So ln(e) = 1 and log₁₀(10) = 1, but ln(10) ≈ 2.3026 and log₁₀(e) ≈ 0.4343. The natural log shows up in growth, decay and calculus because the derivative of ln(x) is simply 1 ÷ x, which is why it is the default in science.

How does log base 2 work, and what is log₂(8)?

log₂ asks how many times you multiply 2 by itself to reach the number. log₂(8) = 3 because 2 × 2 × 2 = 8. log₂(1024) = 10 because 2 to the power 10 is 1024. Binary log is the everyday unit of computer science: it counts bits, the depth of a balanced tree, and the number of halving steps in binary search, so a 1,000,000 element search needs about log₂(1,000,000) ≈ 20 comparisons.

Why must the base be positive and not equal to 1?

A logarithm asks what power turns the base into the argument. If the base were 1, then 1 to any power is always 1, so it could never reach 2 or 10 and the answer would be undefined. A base of zero or a negative base does not give a smooth, single-valued curve over the real numbers. That is why the tool refuses a base of 1, 0 or anything negative and shows a message instead of a wrong number.

What is an antilog and when do I need it?

An antilog reverses a logarithm. If log_b(x) = y, then the antilog gives x = b to the power y. Switch to antilog mode, enter the base and the log value, and the tool returns the original number. For example the antilog of 3 in base 2 is 2 to the power 3 = 8, and the antilog of 2 in base 10 is 100. This is handy for reading off pH, decibels, the Richter scale or any quantity stored on a log scale.

Solve homework logarithms with the steps shown

A worksheet asks for log base 3 of 81 and you need to show your working, not just the answer. Type 81 as x and 3 as the base; the tool returns 4 and prints the change-of-base line ln(81) ÷ ln(3) underneath. You copy the result, copy the steps, and hand in a solution that demonstrates you understand the formula rather than guessing.

Size an algorithm in big-O terms

You are reasoning about a binary search over a million records and want the worst-case comparison count. log₂(1000000) ≈ 19.93, so the search bottoms out in about 20 steps. Switching to log₂ mode and typing the input size gives the depth instantly, which is faster than reaching for a scientific calculator and remembering the change of base by hand.

Read a value off a log scale

pH, decibels and the Richter scale all store numbers on a base-10 log axis. Given a pH reading you need the hydrogen-ion concentration, so you switch to antilog mode, set the base to 10, enter the negative pH value, and read the concentration straight off. The inverse direction is one tab away from the forward log.

Check a spreadsheet or code formula

Your spreadsheet uses LOG(x, base) and you want to confirm a cell before shipping a report. Punch the same x and base into the any-base mode and compare. Because the tool shows ln(x) ÷ ln(b) explicitly, a mismatch tells you whether the bug is your base argument or a typo in the number, not just that two figures disagree.

Logarithm Calculator — log base b, ln, log₁₀, log₂

log_b(x), ln, log₁₀, log₂ and antilog in one place — change-of-base steps shown, one-click copy, browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

What do you want

True number x (x > 0)

Base b (b > 0, b ≠ 1)

Try:

Result

—

Enter the numbers above to see the logarithm.

Every base-b log is computed by change of base: log_b(x) = ln(x) ÷ ln(b). All math runs in your browser, nothing is uploaded.

What this tool does

A focused logarithm calculator that does one job well: it computes log base b of any number, the natural log ln (base e), the common log log₁₀, and the binary log log₂, then shows you the change-of-base step ln(x) ÷ ln(b) behind every answer. Flip to antilog mode and it runs the inverse, recovering the original number from a log value and a base (x equals b raised to y). Inputs are validated the way a textbook defines the domain: the base has to be greater than zero and not equal to one, and the true number has to be greater than zero, so an impossible request returns a clear message instead of a confusing NaN. log₂(8) gives 3, log₁₀(100) gives 2, ln(e) gives 1, and you can read exactly how each result was reached. Everything runs in your browser, the result copies with one click, and the link you share reopens the same calculation.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Logarithm Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Solve homework logarithms with the steps shown
A worksheet asks for log base 3 of 81 and you need to show your working, not just the answer. Type 81 as x and 3 as the base; the tool returns 4 and prints the change-of-base line ln(81) ÷ ln(3) underneath. You copy the result, copy the steps, and hand in a solution that demonstrates you understand the formula rather than guessing.
Size an algorithm in big-O terms
You are reasoning about a binary search over a million records and want the worst-case comparison count. log₂(1000000) ≈ 19.93, so the search bottoms out in about 20 steps. Switching to log₂ mode and typing the input size gives the depth instantly, which is faster than reaching for a scientific calculator and remembering the change of base by hand.
Read a value off a log scale
pH, decibels and the Richter scale all store numbers on a base-10 log axis. Given a pH reading you need the hydrogen-ion concentration, so you switch to antilog mode, set the base to 10, enter the negative pH value, and read the concentration straight off. The inverse direction is one tab away from the forward log.
Check a spreadsheet or code formula
Your spreadsheet uses LOG(x, base) and you want to confirm a cell before shipping a report. Punch the same x and base into the any-base mode and compare. Because the tool shows ln(x) ÷ ln(b) explicitly, a mismatch tells you whether the bug is your base argument or a typo in the number, not just that two figures disagree.

Common pitfalls

Mixing up ln and log₁₀. People type a number into a "log" key expecting natural log, or vice versa. ln has base e ≈ 2.718, log₁₀ has base 10, and ln(10) ≈ 2.3026 is not 1. Pick the mode that names the base explicitly so you never have to guess which one a bare "log" means.
Putting the base and the true number in the wrong slots. log base 2 of 8 is 3, but log base 8 of 2 is one third. The two are reciprocals, not the same. The tool labels x as the true number and b as the base, and the change-of-base line shows ln(x) on top, so a swapped pair is easy to spot.
Feeding a non-positive argument or an illegal base. A logarithm of zero or a negative number is undefined over the reals, and a base of 1, 0 or negative has no logarithm either. Rather than returning NaN, the tool shows a message naming exactly which input broke the domain so you can fix it.

Privacy

Every step — the change-of-base division, the natural, common and binary logs, and the antilog — is plain JavaScript that runs inside your browser tab. No number you type is ever sent to a server and nothing is logged. The one caveat: the shareable link encodes your inputs and mode in the query string, so a link pasted into chat records those values in the recipient server's access log. If a figure is sensitive, use the copy button and paste the result instead of sharing the URL.

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Logarithm Calculator — log base b, ln, log₁₀, log₂

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Logarithm Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Solve homework logarithms with the steps shown

Size an algorithm in big-O terms

Read a value off a log scale

Check a spreadsheet or code formula

Common pitfalls

Privacy

FAQ

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