Logarithms without fear
By MathSolver Editorial Team · Published October 7, 2026 · Updated October 23, 2026
Published October 2026.
A logarithm is a question: log10(1000) asks “10 to what power gives 1000?” Answer: 3. That is the entire concept, everything else is bookkeeping.
The three rules that matter
1. log(ab) = log a + log b, multiplication inside becomes addition outside.
2. log(a/b) = log a − log b, division becomes subtraction.
3. log(an) = n·log a, exponents jump out front. This one solves equations: 2x = 50 becomes x = log 50 ÷ log 2 ≈ 5.64.
Common traps
log(a + b) is not log a + log b, the rules only split multiplication and division. And log of zero or a negative number is undefined in real numbers; if your equation leads there, recheck your algebra. Verify any log computation with our logarithm calculator.
A log is an exponent in disguise
Exponentiation and logarithms undo each other, the same way multiplication and division do. If 2^5 = 32, then log2(32) = 5. It is the same three numbers, rearranged. The base stays the base, and the exponent becomes the answer.
So whenever a log looks intimidating, rewrite it as a power sentence: this base, raised to what power, gives this number? Try it with 10^4 = 10,000. Rearranged, that says log10(10,000) = 4. Or take 3^2 = 9. Rearranged, log3(9) = 2. You just read the same fact backwards.
One special case falls straight out of this idea. Any base raised to the power 1 is itself, so the log of the base is always 1: log10(10) = 1 and log2(2) = 1.
The three logs you will actually meet
| Name | Written as | Base | Example |
|---|---|---|---|
| Common log | log | 10 | log10(100) = 2, because 10^2 = 100 |
| Natural log | ln | e (about 2.718) | ln(e^3) = 3 |
| Binary log | log2 | 2 | log2(32) = 5, because 2^5 = 32 |
The common log is the default on most calculators, where the button just says LOG. The natural log, ln, shows up wherever things grow or decay continuously, like interest and population models. The binary log belongs to computing: each doubling of data adds exactly 1 to log2, so log2(1024) = 10 tells you ten doublings get you past a thousand.
Watching the three rules work
The rules from earlier are easier to trust once you see them check out with real numbers.
Product rule: log10(100 × 1000) should equal log10(100) + log10(1000) = 2 + 3 = 5. Direct check: 100 × 1000 = 100,000, which is 10^5, so log10(100,000) = 5. It matches.
Quotient rule: log2(32 / 4) should equal log2(32) – log2(4) = 5 – 2 = 3. Direct check: 32 / 4 = 8, and 2^3 = 8, so log2(8) = 3. It matches again.
Power rule: log2(8^2) should equal 2 × log2(8) = 2 × 3 = 6. Direct check: 8^2 = 64, and 2^6 = 64, so log2(64) = 6. Three rules, three exact matches. When a rule feels abstract, test it on small numbers like these first.
Logs in the wild: the pH scale
Chemists measure acidity with pH, defined as the negative of the base 10 log of the hydrogen ion concentration. Pure water has a concentration of 10^-7, so its pH is 7. Lemon juice sits near 10^-2, so its pH is 2.
The log is doing a job here. Concentrations range over many powers of ten, too wide for a normal scale. Because pH is logarithmic, each single step means a tenfold change: a liquid at pH 3 has 10 times the hydrogen ion concentration of one at pH 4. Sound engineers use the same trick with decibels, where multiplying intensity by 10 adds exactly 10 dB.
Mistakes that trip everyone once
1. Reading a log as multiplication. log2(8) is not 2 times 8, and it is not 4 just because 2 × 4 = 8. It asks for a power: 2^3 = 8, so log2(8) = 3.
2. Mixing up the ln and log buttons. They answer different questions. log10(100) = 2, but ln(100) is about 4.605, because e needs a bigger power than 10 to reach 100. One wrong button press ruins the rest.
3. Mixing bases inside the rules. The product, quotient and power rules only work when every log in the line shares the same base. log2(8) + log2(4) = 3 + 2 = 5 is fine. Swap one to base 10 and the rule no longer applies.
Frequently asked questions
Why is log(1) always 0?
Because every base raised to the power 0 equals 1, and the log asks which power produces the input. For the input 1, that power is always 0.
What does a log with no base written mean?
In school and most science, a plain log with no subscript means base 10. In advanced math and some programming, it often means base e instead. When it matters, the base is written as a small subscript, like the 2 in log2. If unsure, test both readings and see which gives a sensible answer.
Can a logarithm be negative?
Yes. Inputs between 0 and 1 have negative logs. log10(0.01) = -2, because 10^-2 = 0.01. Only zero and negative inputs are off limits in real numbers.
What is e, and why does it get its own log?
e is a constant, about 2.718, that appears whenever growth is smooth and continuous. The natural log, ln, is simply log base e. It earns a special name because calculus treats it as the most natural log of all: the derivative of ln(x) is the tidy 1/x.