This antilog calculator helps you quickly find the antilogarithm of any number with any base — whether you're working with base 10, natural logarithms (base e), binary (base 2), or any custom base. Simply enter your logarithm value, choose your base, and the result appears instantly.
Whether you're a student working through a logarithms chapter, a scientist converting data from a logarithmic scale, or an engineer who just needs a quick answer without reaching for a scientific calculator, this tool takes the friction out of antilog calculations. It handles positive values, negative values, decimals, and any base you throw at it.
What Is an Antilogarithm?
If you've worked with logarithms, you already understand antilogs — you just might not know it yet. An antilogarithm simply reverses what a logarithm does.
A logarithm asks: "What power do I raise this base to in order to get a certain number?" An antilog flips that around and asks: "If I raise the base to this power, what number do I get?"
Here's a quick way to see it. You probably know that 10 raised to the 3rd power equals 1,000. That means:
- Log direction: log₁₀(1,000) = 3 — "10 to what power gives me 1,000? Answer: 3."
- Antilog direction: antilog₁₀(3) = 1,000 — "10 raised to the 3rd power gives me what? Answer: 1,000."
Same relationship, just read in the opposite direction. In mathematical notation:
If log_b(y) = x, then antilog_b(x) = y
Once you see antilogs as simply "going backwards" through a logarithm, they stop being intimidating.
The Antilog Formula
The formula for calculating an antilogarithm is refreshingly simple:
antilog_b(x) = b^x
That's it — you raise the base (b) to the power of your logarithm value (x), and you have your answer.
Here's how this works for the three bases you'll encounter most often:
Base | Name | Formula | Quick Example |
|---|---|---|---|
10 | Common antilog | 10^x | antilog₁₀(2) = 10² = 100 |
e ≈ 2.718 | Natural antilog | e^x | antilog_e(1) = e¹ ≈ 2.718 |
2 | Binary antilog | 2^x | antilog₂(8) = 2⁸ = 256 |
You'll run into the common antilog (base 10) most frequently in chemistry, physics, and general math courses. The natural antilog (base e) shows up in calculus, statistics, and anything involving continuous growth. And base 2 is the backbone of computing — memory sizes, data structures, and algorithm analysis all rely on powers of 2.
How to Use This Calculator
Using this calculator takes about five seconds:
- Enter the logarithm value. This is the exponent — the number you want to "undo" the log for. For instance, type "2" if you need the antilog of 2.
- Set the base. The default is 10, which works for common logarithms. Change it to 2.71828 for natural logs, 2 for binary, or any other positive number your problem requires.
- Read your result. The antilog appears instantly below. With a log value of 2 and base 10, you'll see 100 — because 10² = 100.
No formulas to remember, no antilog tables to look up, no fiddling with calculator buttons. Just enter, read, and move on.
Antilog Reference Table (Base 10)
This table covers the common antilog values you'll encounter most often in math and science. It's worth spending a moment noticing the pattern:
Logarithm Value (x) | Antilog₁₀(x) = 10^x |
|---|---|
-3 | 0.001 |
-2 | 0.01 |
-1 | 0.1 |
0 | 1 |
0.5 | 3.162 |
1 | 10 |
1.5 | 31.623 |
2 | 100 |
2.5 | 316.228 |
3 | 1,000 |
4 | 10,000 |
5 | 100,000 |
Notice how each whole-number increase in the logarithm value multiplies the result by 10. Go from 1 to 2, and you jump from 10 to 100. Go from 2 to 3, and you jump from 100 to 1,000. That consistent tenfold scaling is exactly why logarithmic scales are so useful for handling numbers that span huge ranges.
And if you're wondering about the negative values — don't worry, they're perfectly normal. A negative logarithm value just means the original number is between 0 and 1. The antilog of -1 is 0.1, the antilog of -2 is 0.01, and so on.
Worked Examples
Reversing a Common Logarithm
Say you're working through a textbook problem and you've arrived at log₁₀(x) = 3.5. You need to find x.
antilog₁₀(3.5) = 10^3.5
You can break this down: 10³ is 1,000 and 10^0.5 is about 3.162, so 10^3.5 ≈ 3,162.28. Or just plug 3.5 into this calculator with base 10 and skip the mental math entirely.
Finding Hydrogen Ion Concentration from pH
This is one of the most common real-world antilog calculations. If a solution has a pH of 5.2, what's the hydrogen ion concentration?
Since pH = -log₁₀[H⁺], you reverse it: [H⁺] = 10^(-pH) = 10^(-5.2) ≈ 6.31 × 10⁻⁶ mol/L
Enter -5.2 as the logarithm value with base 10. Chemistry students run into this conversion constantly — having a quick calculator on hand saves a lot of time during problem sets.
Working with Natural Logarithms
A population growth model tells you that ln(P) = 4.6. What's the actual population value P?
P = e^4.6 ≈ 99.48
To use this calculator for natural logs, enter 4.6 as the logarithm value and set the base to 2.71828 (Euler's number).
Computing Memory Addresses (Base 2)
In computer science, a system with 16 address bits can reference 2^16 unique memory locations. How many is that?
antilog₂(16) = 2^16 = 65,536 addresses
Enter 16 as the logarithm value and 2 as the base. This kind of calculation comes up whenever you're working with bit widths, binary trees, or hash table sizes.
Using a Custom Base
If log₅(x) = 3, what is x?
antilog₅(3) = 5³ = 125
Enter 3 as the logarithm value and 5 as the base. This calculator handles any positive base, so you're not limited to the standard ones.
Where Antilogs Show Up in Real Life
You might think of antilogs as a purely academic exercise, but they quietly power a lot of everyday measurements and systems:
Chemistry and pH. Every time a chemist or biology student converts a pH reading into an actual hydrogen ion concentration, they're calculating an antilog. The pH scale is logarithmic by design — a pH of 3 doesn't mean "three times as acidic" as pH 1; it means the concentration differs by a factor of antilog₁₀(2) = 100.
Sound and decibels. When an audio engineer says a signal is 30 dB louder, that corresponds to a 1,000× increase in power — because antilog₁₀(3) = 1,000. The decibel scale compresses enormous ranges of sound intensity into manageable numbers, and antilogs translate those numbers back into physical reality.
Earthquakes. The Richter scale works the same way. A magnitude 6 earthquake has ground motion 100 times greater than a magnitude 4, not twice as much. That's antilog₁₀(2) = 100 at work.
Finance and growth modeling. Exponential growth models — for investments, populations, and biological processes — often use natural logarithms. Recovering the actual dollar value or population count from log-transformed data means computing e^x, the natural antilog.
Computer science. Binary antilogs tell you concrete quantities: how many nodes a binary tree of depth n holds (2^n), how many values an n-bit integer can represent, or the maximum size of a data structure at a given depth.
Tips for Working with Antilogs
Match your base to the notation in your problem. This is the most common source of errors. If you see "ln" anywhere, that means natural logarithm — use base e (≈ 2.71828). If you see "log" without a subscript, it almost always means base 10 in science and engineering courses, though some pure math texts default to base e. When in doubt, check the conventions your course or textbook uses.
Negative inputs aren't a problem. A negative logarithm value simply means your answer will be a number between 0 and 1. For example, antilog₁₀(-2) = 0.01. This comes up often in chemistry (pH values above 7 involve negative logs of concentration) and signal processing (negative decibel values indicate attenuation).
Use antilogs to double-check your work. If you calculated that log₁₀(500) ≈ 2.699, verify it by computing antilog₁₀(2.699). If the result is close to 500, your original calculation checks out. Logarithms and antilogarithms undo each other — antilog_b(log_b(x)) = x — so they make a natural error-checking pair.
Keep the relationship in your head. The core identity is simple: a logarithm finds the exponent, and an antilog raises the base to that exponent to get back the original number. Once that clicks, you'll find antilogs intuitive rather than abstract.