A logarithm is the inverse of an exponent. It answers the question: what power do I raise this base to in order to get this number? If 10² = 100, then log₁₀(100) = 2. The logarithm undoes the exponentiation. Subtraction undoes addition, division undoes multiplication, and logarithms undo exponents.

What does log(100) equal?

log(100) = 2, because 10² = 100. When you see 'log' without a subscript, it means log base 10 (common logarithm). The question log(100) is asking: 10 to what power equals 100? The answer is 2, because 10 × 10 = 100.

What is the difference between log and ln?

log means logarithm base 10. ln means logarithm base e (approximately 2.71828), called the natural logarithm. They're the same operation with different bases. log(100) = 2 because 10² = 100. ln(e) = 1 because e¹ = e. Most scientific calculators have both buttons — LOG for base 10, LN for base e.

Why do calculators use base 10 and base e instead of other bases?

Base 10 (common log) aligns with our decimal number system and makes it easy to work with powers of 10, orders of magnitude, and scientific notation. Base e (natural log) appears constantly in calculus, compound interest, exponential growth, and decay problems because the derivative of eˣ is eˣ. Those two bases cover nearly every practical application.

How do you calculate a logarithm without a calculator?

For simple powers of 10, count the zeros or the exponent. log(1000) = 3 because 10³ = 1000. For powers of 2, count the doublings. log₂(32) = 5 because 2⁵ = 32. For anything else, you need a calculator, logarithm tables (historical), or iterative approximation methods. Logarithms of non-integer powers don't resolve to neat whole numbers.

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Logarithms in 5 Minutes: The Inverse of Exponents

Logarithms are the inverse of exponents. That's the entire concept. Everything else — the base-10 log button on your calculator, the natural log that shows up in calculus and compound interest, the properties that let you turn multiplication into addition — flows from that one fact. If you understand exponents, you already understand logarithms. Here's the proof in five minutes.

Last updated: July 29, 2026.

The core idea: logarithms undo exponents

You already know that subtraction undoes addition and division undoes multiplication. Logarithms undo exponentiation the same way.

If 10² = 100, then log₁₀(100) = 2. The logarithm answers the question: what power do I raise 10 to in order to get 100? The answer is 2. That's what log₁₀(100) means.

General form:

If b^x = y, then log_b(y) = x

The base b is the number you're exponentiating. The result y is what you get. The logarithm gives you back the exponent x. It's a reverse lookup.

The two logarithms you'll actually use

There are infinitely many possible bases, but in practice you only need two:

log (base 10): called the common logarithm. Written as log(x) or log₁₀(x). This is the LOG button on your calculator.
ln (base e): called the natural logarithm. Written as ln(x) or logₑ(x). This is the LN button on your calculator. The base e is Euler's number, approximately 2.71828.

When you see "log" with no subscript, it almost always means base 10. When you see "ln," it always means base e. Those are the defaults. Everything else — log₂, log₅, whatever — you specify explicitly.

Worked examples: base 10

Base 10 is the easiest to internalize because our number system is base 10. Powers of 10 are just 1 followed by zeros.

10¹ = 10, so log(10) = 1
10² = 100, so log(100) = 2
10³ = 1,000, so log(1000) = 3
10⁰ = 1, so log(1) = 0
10⁻¹ = 0.1, so log(0.1) = −1

The logarithm is just counting the exponent. log(1000) asks: 10 to what power equals 1000? The answer is 3, because 10 × 10 × 10 = 1000. If you can count zeros, you can do base-10 logarithms of powers of ten in your head.

What about numbers that aren't neat powers of 10? They fall between the integers. log(50) is somewhere between 1 and 2, because 50 is between 10¹ and 10². Specifically, log(50) ≈ 1.699, because 10^1.699 ≈ 50. You need a calculator for that.

Worked examples: natural log (base e)

The natural logarithm uses base e, which is an irrational number like π. You don't need to memorize e's digits (2.71828...), just that it's the base of the natural log and that it shows up constantly in calculus and exponential growth.

e¹ = e, so ln(e) = 1
e² ≈ 7.389, so ln(7.389) ≈ 2
e⁰ = 1, so ln(1) = 0
e⁻¹ ≈ 0.368, so ln(0.368) ≈ −1

The natural log is harder to do in your head because e isn't a round number. ln(10) ≈ 2.303, ln(100) ≈ 4.605, ln(1000) ≈ 6.908. You press the LN button and trust the calculator.

Why base e matters: it's the calculus-friendly base

Base 10 is convenient for humans. Base e is convenient for math.

The reason: the derivative of e^x is e^x. It's the only exponential function that doesn't change when you differentiate it. That one property makes e^x and ln(x) show up everywhere in calculus, physics, probability, compound interest, population growth, radioactive decay — anything involving continuous change over time.

When you see a formula like A = Pe^rt for continuously compounded interest, the e is there because the rate of growth is proportional to the current amount at every instant. Base 10 doesn't have that property. Base 2 doesn't either. Only e.

You don't need to understand why that works to use logarithms. You just need to know: if the problem involves continuous growth, decay, or rates of change, you're probably using ln. If it's about orders of magnitude, scientific notation, or powers of 10, you're using log.

The logarithm properties (and why they matter)

Logarithms have three core properties that make them useful for simplifying calculations. These work for any base.

1. Product rule: log(a × b) = log(a) + log(b)

Multiplication becomes addition. Before calculators, this was huge — you could multiply large numbers by adding their logarithms, then converting back. Example: log(10 × 100) = log(10) + log(100) = 1 + 2 = 3, and 10³ = 1000. Same answer, easier arithmetic.

2. Quotient rule: log(a / b) = log(a) − log(b)

Division becomes subtraction. Example: log(1000 / 10) = log(1000) − log(10) = 3 − 1 = 2, and 10² = 100. Check: 1000 / 10 = 100. It works.

3. Power rule: log(a^b) = b × log(a)

Exponents become multiplication. Example: log(10³) = 3 × log(10) = 3 × 1 = 3. This is the property that makes solving exponential equations possible — you can "bring down" the exponent and isolate the variable.

These rules are why slide rules worked. Engineers would add and subtract distances on a logarithmic scale to multiply and divide numbers mechanically. Logarithms turned hard arithmetic into easy geometry.

When you actually need logarithms

Most people see logarithms in algebra class, never use them again, and assume they're theoretical. They're not. Here are the real-world cases where you need them:

Solving for time in compound interest: If you want to know how long it takes to double your money at 6% annual interest, you solve 2 = (1.06)^t by taking the log of both sides: log(2) = t × log(1.06), so t = log(2) / log(1.06) ≈ 11.9 years.
Exponential decay: Half-life calculations, drug dosages, radioactive dating. If you know the decay rate and want to find the time, you need ln.
Scientific notation and orders of magnitude: log₁₀ tells you how many digits a number has. log(1,000,000) = 6, which is why a million is a 7-digit number (10⁶ = 1,000,000).
pH, decibels, Richter scale: All logarithmic scales. A pH of 7 is 10 times less acidic than pH 6, which is 10 times less acidic than pH 5. Each step is a power of 10, which is why the scale uses log₁₀.
Information theory and computer science: log₂ measures bits. If you have 8 possibilities, you need log₂(8) = 3 bits to encode them. Binary search runs in O(log n) time because you halve the problem at each step.

If the unknown is in the exponent, you need logarithms to get it out. That's the use case. Any equation that looks like a^x = b gets solved by taking the log of both sides.

How to calculate logarithms (the practical way)

For anything that isn't a clean power of the base, you use a calculator. Modern calculators and scientific computing environments have LOG (base 10) and LN (base e) built in. Press the button, type the number, done.

For other bases, use the change of base formula:

log_b(x) = log(x) / log(b)

or equivalently log_b(x) = ln(x) / ln(b).

Example: to find log₂(32), calculate log(32) / log(2) ≈ 1.505 / 0.301 ≈ 5. Check: 2⁵ = 32. Correct.

You can use the scientific calculator for both log and ln. The LOG button is base 10, the LN button is base e. For other bases, use the change-of-base formula and divide.

Common mistakes and how to avoid them

Mistake 1: Thinking log(a + b) = log(a) + log(b). This is false. log(10 + 100) ≠ log(10) + log(100). The left side is log(110) ≈ 2.041. The right side is 1 + 2 = 3. Not the same. Logarithms turn multiplication into addition, not addition into addition.

Mistake 2: Confusing log and ln. They're different bases. log(100) = 2, but ln(100) ≈ 4.605. If a formula says ln, don't use LOG. If it says log, don't use LN. The answers won't match.

Mistake 3: Forgetting that log(1) = 0 for any base. This trips people up in algebra. If you see log(x) = 0, it means x = 1, because 10⁰ = 1 (or e⁰ = 1, depending on the base). The logarithm of 1 is always zero.

Mistake 4: Trying to take the log of a negative number or zero. Logarithms are only defined for positive real numbers. log(0) is undefined (you can't raise a positive base to any power and get zero). log(−5) is undefined in the real numbers (though it exists in complex numbers, which is a different topic). If your calculator shows ERROR, check whether you're taking the log of something ≤ 0.

A worked example: doubling time

How long does it take for an investment to double at 8% annual interest, compounded annually?

Set up the equation. You start with principal P. After t years at 8%, you have P(1.08)^t. You want that to equal 2P (doubled). So:

P(1.08)^t = 2P

Divide both sides by P:

(1.08)^t = 2

Take the logarithm of both sides (either log or ln works; we'll use log):

log((1.08)^t) = log(2)

Use the power rule to bring down the exponent:

t × log(1.08) = log(2)

Solve for t:

t = log(2) / log(1.08)

Plug into a calculator:

t ≈ 0.3010 / 0.0334 ≈ 9.0 years

Your money doubles in roughly 9 years at 8% annual interest. That's the kind of problem logarithms were invented to solve — you can't isolate t without them, because it's trapped in the exponent.

Frequently asked questions

Can I calculate logarithms in my head?

Only for clean powers of the base. log(1000) = 3 because you can count the zeros (or recognize 10³). log₂(64) = 6 because you can count the doublings (2 → 4 → 8 → 16 → 32 → 64). For anything else — log(50), ln(7), log₂(100) — you need a calculator or logarithm tables. There's no mental shortcut that gets you the exact answer.

Why does my calculator only have LOG and LN buttons?

Because those two bases (10 and e) cover 99% of real-world use cases, and you can convert any other base using the change-of-base formula. If you need log₂(x), calculate log(x) / log(2) or ln(x) / ln(2). Same answer, one extra division. There's no need to dedicate a button to every possible base.

What's the relationship between log and ln?

They're the same operation with different bases. The conversion is ln(x) = log(x) × ln(10), or log(x) = ln(x) / ln(10). Since ln(10) ≈ 2.303, you can approximate: ln(x) ≈ 2.3 × log(x). Example: log(100) = 2, so ln(100) ≈ 2.3 × 2 = 4.6. Actual value: ln(100) ≈ 4.605. Close enough for estimation.

Why is e called the natural base?

Because it's the base that makes calculus work cleanly. The derivative of e^x is e^x, and the derivative of ln(x) is 1/x. No other base has those properties without tacked-on correction factors. When you model continuous growth, compounding interest at every instant, or anything involving rates proportional to the current value, e is the base that falls out of the math naturally — hence "natural logarithm."

Can logarithms be negative?

Yes. log(0.1) = −1 because 10⁻¹ = 0.1. Any number between 0 and 1 has a negative logarithm. As the input approaches zero from the right, the logarithm goes to −∞. As the input grows, the logarithm increases without bound. The logarithm itself can be any real number; it's the input that must be positive.

What's the log of zero?

Undefined. There is no exponent you can raise a positive base to and get zero. 10⁻¹⁰⁰ is a very small number but still positive. As the input to log(x) approaches zero from the right, the output goes to −∞, but log(0) itself doesn't exist. Same for log of a negative number — undefined in the real numbers (though defined in complex analysis, where it picks up an imaginary component).

Try it yourself

Pick an exponential equation you care about — maybe a loan balance growing with interest, or a population doubling every few years — and solve for the unknown time or rate using logarithms. The first time you take the log of both sides and the exponent comes down as a coefficient, it clicks. You stop seeing logarithms as abstract and start seeing them as the tool that unsticks exponential algebra.

This article is for general education. Logarithms are a mathematical tool, not financial advice.