What a Logarithm Actually Asks
A logarithm is a question about exponents: “which power of the base gives this number?” Written out, logb(x) = y means by = x. So log2(32) is asking “2 to what power makes 32?” and the answer is 5.
log₂(32)
- Rewrite the question as 2? = 32.
- List powers of 2: 2, 4, 8, 16, 32.
- 32 is the fifth entry, so 25 = 32.
- Therefore the exponent, and the logarithm, is 5.
Answer: 5
John Napier published the first log tables in 1614 so astronomers could turn long multiplications into additions. Slide rules ran on that idea for about 350 years, which is why the log laws below are all about turning one operation into an easier one.
Log, Ln and Log Base 2: Which One Is Which?
Three bases cover nearly every problem you will meet. The key labelled log on a calculator is base 10, ln is base e ≈ 2.71828, and log2 is the one computer scientists use for bits.
| Name | Base | Where it shows up | Example |
|---|---|---|---|
| log (common) | 10 | pH, decibels, Richter scale | log 1000 = 3 |
| ln (natural) | e ≈ 2.71828 | Growth, decay, calculus, interest | ln e² = 2 |
| log2 (binary) | 2 | Bits, binary search, music octaves | log2 1024 = 10 |
If your calculator has no log2 key, use change of base: logb(x) = ln x ÷ ln b. That is exactly what this page does when you pick any base other than 10 or e.
The Three Log Laws and Change of Base
| Law | Rule | Example |
|---|---|---|
| Product | logb(xy) = logbx + logby | log 6 = log 2 + log 3 |
| Quotient | logb(x/y) = logbx − logby | log(5/2) = log 5 − log 2 |
| Power | logb(xn) = n·logbx | log 8 = 3·log 2 |
| Change of base | logbx = ln x ÷ ln b | log210 ≈ 3.3219 |
log₃(54) − log₃(2)
- Both logs share base 3, so the quotient law applies.
- log₃(54) − log₃(2) = log₃(54 ÷ 2).
- 54 ÷ 2 = 27, so we need log₃(27).
- 33 = 27, so log₃(27) = 3.
Answer: 3
One warning: there is no law for the log of a sum. log(a + b) is not log a + log b, and it is the single most common slip on tests.
How to Solve a Logarithmic Equation
The plan never changes: combine the logs into one, rewrite in exponential form, solve, then check the domain. The last step matters because combining logs can quietly widen the set of allowed values.
log₂(x) + log₂(x − 2) = 3
- Combine with the product law: log₂(x(x − 2)) = 3.
- Write it as a power: x(x − 2) = 23 = 8.
- Expand and move everything left: x² − 2x − 8 = 0, which factors as (x − 4)(x + 2) = 0.
- Candidates are x = 4 and x = −2.
- The original needs x > 0 and x − 2 > 0, so x = −2 is rejected. Check x = 4: log₂4 + log₂2 = 2 + 1 = 3 ✓
Answer: x = 4
Solving Exponential Equations With Logarithms
When the unknown sits in an exponent, take a log of both sides. The power law then drops the exponent down where you can reach it. This is how you find how long money takes to double, or how old a bone is from its carbon-14.
5x = 20
- Take ln of both sides: ln(5x) = ln 20.
- Use the power law: x·ln 5 = ln 20.
- Divide: x = ln 20 ÷ ln 5.
- ln 20 ≈ 2.99573 and ln 5 ≈ 1.60944, so x ≈ 1.8614.
Answer: x ≈ 1.8614
The same move with 1.05t = 2 gives t = ln 2 ÷ ln 1.05 ≈ 14.2, so money growing at 5% a year doubles in a little over fourteen years.
Real Numbers: Logs You Can Check
| Expression | Value | Why it matters |
|---|---|---|
| log₁₀ 2 | 0.30103 | Each doubling adds about 0.3 to a base-10 log |
| ln 2 | 0.693147 | Doubling time = ln 2 ÷ ln(1 + rate) |
| log₂ 1024 | 10 | 1,024 = 2¹⁰, the binary kilobyte |
| 10^1.5 | 31.62 | Each +1 on the Richter scale is 10× the amplitude, about 31.6× the energy |
The “rule of 72” is a log shortcut. At 7% growth the exact doubling time is ln 2 ÷ ln 1.07 = 10.245 years; the rule gives 72 ÷ 7 = 10.29. It is close because ln 2 ≈ 0.693 and ln(1 + r) ≈ r for small r.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising every log rule as a separate fact. They are the exponent rules read backwards: log(ab) = log a + log b is just aᵐ × aⁿ = aᵐ⁺ⁿ.
Underrated: checking by raising the base. If log₃ 81 = 4 then 3⁴ must be 81. It takes five seconds and catches every sign slip.
Plain wrong: splitting log(a + b) into log a + log b. There is no such rule. log(2 + 3) = 0.699, while log 2 + log 3 = 0.778.
Logarithm Mistakes That Cost Marks
- The input of a log must be positive. log(0) and log(−5) are undefined for real numbers.
- The base must be positive and not 1. log1 has no meaning because 1 to any power is 1.
- Writing log(x + y) = log x + log y. No such law exists.
- Forgetting to check answers after combining logs. Extraneous roots are the usual trap.
- Typing log when you mean ln. They differ by a factor of about 2.3026.
Why Students Get Stuck on Logarithms, and What a Good Calculator Should Do
Searching for help with logarithms usually starts late at night with one problem that will not work out. The rules look alike, and log and ln are easy to mix up. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to use the log laws and change of base without slips, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Writing log(a + b) = log a + log bSolve log(x) + log(x - 3) = 1↗
- Mixing up log and lnEvaluate log base 2 of 32↗
- Forgetting that the argument must be positiveSolve 2^x = 20↗
- Not knowing how to change baseSimplify ln(e^3) + ln(1)↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants each log law named
- Wants equations with logs solved and checked
- Wants the base-change formula applied
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Solve log(x) + log(x - 3) = 1↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.