Log Calculator
A log calculator answers one question: to what power must the base be raised to get this number. Log base 10 of 100 is 2, because 10² = 100, and log base 10 of 1,000 is 3, because 10³ = 1,000. The natural log, written ln, uses base e, so ln(10) ≈ 2.303, and log base 2 of 32 is 5, because 2⁵ = 32. The scientific calculator above evaluates all three, and this page explains the log rules that make them useful.
Logarithms turn multiplication into addition, which is why they appear in science, finance, and computing: the Richter scale, the pH scale, decibels, and compound interest all run on logs. The rules are short: log(a × b) = log(a) + log(b), log(a ÷ b) = log(a) − log(b), and log(aⁿ) = n × log(a). This page works through the common values, shows the three bases, and explains how to change from one base to another.
Enter an expression above, or tap the keys. Examples: sin(30), 2^10, sqrt(144), 5!, log(1000), ln(e).
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How a Scientific Calculator Works: PEMDAS and Functions
A scientific calculator differs from a basic one in one decisive way: it follows the order of operations, known as PEMDAS — Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). This is the rule that makes 2 + 3 × 4 equal 14, not 20, because the multiplication happens before the addition. Parentheses override the order: (2 + 3) × 4 = 20, because the bracketed addition happens first. Every expression on this page is evaluated with PEMDAS, so what you type is what mathematics means, not what a pocket calculator would do if it simply worked left to right.
Beyond arithmetic, a scientific calculator provides powers, roots, logarithms, trigonometry, and factorials. The expression 2^10 = 1024, sin(30°) = 0.5, log10(1000) = 3, and 5! = 120 are all one-line jobs. Angles need care: trigonometric functions depend on whether the calculator is in degree mode or radian mode, and sin(30°) = 0.5 only in degrees, while in radians sin(30) ≈ −0.99. The calculator above shows its mode clearly, and you can switch it before evaluating any angle.
The National Institute of Standards and Technology maintains the reference values for the mathematical constants and functions these calculators implement, from pi to the natural logarithm. When a result looks surprising, the first things to check are the mode, the parentheses, and the order of operations — those three explain nearly every “wrong answer” a scientific calculator ever gives.
Step 1 — Type the Expression as You Would Write It
Enter the whole expression, including parentheses, in the display: 2+3*4, or (2+3)*4 when you mean the addition first. The calculator reads the expression as written, so the parentheses you include are the parentheses it obeys.
Step 2 — Check the Angle Mode
Before any sine, cosine, or tangent, confirm DEG or RAD. School and most practical problems use degrees; calculus and physics formulas usually use radians. A wrong mode is the single most common cause of a wrong trigonometry answer.
Step 3 — Evaluate, Then Check the Answer
Press equals and read the result, then sanity-check it: 2 + 3 × 4 must be 14, sin(30°) must be 0.5, and log10(1000) must be 3. If a familiar value comes out wrong, the mode or a missing parenthesis is the culprit, not the calculator.
Worked Examples: Logarithms
Each row shows a log, the power it represents, and the check by raising the base to the answer.
| Logarithm | Meaning | Value |
|---|---|---|
| log10(100) | 10 to what power is 100 | 2 |
| log10(1000) | 10 to what power is 1,000 | 3 |
| ln(10) | e to what power is 10 | 2.303 |
| log2(32) | 2 to what power is 32 | 5 |
| log10(50) | 10 to what power is 50 | 1.699 |
| ln(e²) | e to what power is e² | 2 |
The value log10(50) ≈ 1.699 shows why logs are not always whole numbers: 50 sits between 10 and 100, so its base-10 log sits between 1 and 2, and 10^1.699 = 50.00 confirms it. The natural log of e² is exactly 2, because the natural log undoes the exponential: ln(eⁿ) = n for any n. Try each value in the calculator above, using the log key for base 10 and the ln key for base e, and confirm the results.
How to Use the Scientific Calculator
Type an expression or click the buttons, press equals, and the result appears with a running history of your calculations. It handles parentheses, powers, roots, logs, trigonometry, and factorials, and it remembers your last answers so you can reuse them.
- Type or click the expression. Use the keyboard or the on-screen keys; both feed the same display, and the history below records each calculation.
- Use parentheses for grouping. Anything inside parentheses is evaluated first, so (2+3)*4 gives 20 while 2+3*4 gives 14, exactly as PEMDAS requires.
- Switch DEG and RAD for angles. Tap the mode button before trigonometry: sin(30) is 0.5 in degrees and about −0.99 in radians.
- Use the function keys. The buttons for powers, square roots, logarithms, and factorials insert the function at the cursor, so 5! evaluates to 120 and log(1000) to 3.
- Reuse answers from history. Click a line in the history to load that expression back into the display and edit it, instead of retyping the whole thing.
The Three Bases, Compared
Base 10, base e, and base 2 each have a home: base 10 in measurement scales, base e in growth and decay, and base 2 in computing. This table compares them on the same numbers.
| Number | log10 | ln (base e) | log2 |
|---|---|---|---|
| 10 | 1 | 2.303 | 3.322 |
| 100 | 2 | 4.605 | 6.644 |
| 1,000 | 3 | 6.908 | 9.966 |
| 32 | 1.505 | 3.466 | 5 |
| 2 | 0.301 | 0.693 | 1 |
Common Mistakes With Scientific Calculators
Most scientific calculator errors are not arithmetic errors: they are order, mode, and parenthesis errors. Check these first.
- Ignoring PEMDAS. Typing 2+3*4 and expecting 20 is the classic error: multiplication comes before addition, so the answer is 14. Use parentheses when you mean the addition first.
- Wrong angle mode. Degrees and radians give different answers for the same angle number. Always check the mode before trigonometry, and after switching.
- Unbalanced parentheses. A missing closing parenthesis makes the whole expression mean something else, or nothing at all. Count the opening and closing brackets before pressing equals.
- Forgetting that log means base 10. On most calculators, log is base 10 and ln is base e. log(100) = 2, while ln(100) ≈ 4.61, and mixing them up is a frequent exam error.
- Trusting a rounded intermediate. Copying a rounded result such as 0.58 into the next step, instead of using the full stored value, adds error at every step. Keep full precision until the end.
Where You Will Use a Scientific Calculator
Any problem with powers, roots, angles, or logarithms needs a scientific calculator, and those problems appear in school, work, and home projects alike.
- School maths and science. Algebra, geometry, trigonometry, and physics homework all rely on powers, roots, and trig functions that a basic calculator cannot do.
- Engineering and trades. Electricians, builders, and engineers compute angles, loads, and conversions that need sines, cosines, and exponents.
- Finance. Compound interest, loan growth, and investment returns are exponential, and a scientific calculator handles the powers directly.
- Navigation and surveying. Bearings, distances, and triangle solutions are trigonometry problems, solved with sine and cosine.
- Everyday problem solving. Splitting a bill with a tip and tax, scaling a recipe by a fractional factor, or checking a square corner with Pythagoras all benefit from one careful evaluation.
The Log Rules, With Numbers
The log rules are the reason logs are useful, and each one is easy to verify with numbers. The product rule says log(10 × 100) = log(10) + log(100) = 1 + 2 = 3, and indeed log(1,000) = 3. The quotient rule says log(1,000 ÷ 10) = log(1,000) − log(10) = 3 − 1 = 2, and indeed log(100) = 2. The power rule says log(10³) = 3 × log(10) = 3 × 1 = 3. These rules turn hard multiplications into easy additions, which is exactly what slide rules did for two centuries before electronic calculators.
Tips for Accurate Results
Good calculator habits are mostly about checking, not typing. These five keep your answers honest.
- Work out the rough answer first. A quick mental estimate tells you whether the calculator’s result is in the right range before you trust it.
- Parenthesise generously. Extra parentheses never change a correct expression, and they prevent the most common order-of-operations mistakes.
- Confirm the mode every session. The mode can persist from an earlier problem, so check DEG or RAD before your first trigonometry calculation, not after.
- Use the history. The history line shows exactly what was evaluated, so a wrong answer can be traced to the exact expression that produced it.
- Keep full precision mid-problem. Round only the final answer; intermediate rounding is the quietest way to lose marks and money.
Scientific Calculator Quick Answers
These are the expressions people evaluate most often, each one verified against the calculator above.
| Expression | Result |
|---|---|
| 2 + 3 × 4 | 14 |
| (2 + 3) × 4 | 20 |
| 2^10 | 1,024 |
| sin(30°) | 0.5 |
| log10(1000) | 3 |
Related tools on this site: the main scientific calculator, basic calculator, square root calculator, percentage calculator, and the scientific calculator again whenever you need a fresh check.
Frequently Asked Questions
What is log10(100)?
Log base 10 of 100 is 2, because 10² = 100. The base-10 log asks to what power 10 must be raised to get the number, and 10 raised to 2 is 100.
What is the natural log of 10?
The natural log of 10, written ln(10), is about 2.303. The natural log uses base e, which is about 2.71828, and e^2.303 = 10.00 confirms the value.
What is log2(32)?
Log base 2 of 32 is 5, because 2⁵ = 32. Base-2 logs count doublings, which is why they appear in computing: 32 is five doublings from 1.
How do I change log bases?
Use the change-of-base formula: log base b of x equals ln(x) ÷ ln(b), or log10(x) ÷ log10(b). For log2(32), that is ln(32) ÷ ln(2) = 3.466 ÷ 0.693 = 5.
Why are logarithms used for earthquakes and pH?
Because the quantities span huge ranges, and a log scale compresses them. Each whole step on the Richter scale is ten times the amplitude, and each pH unit is ten times the acidity, so the log turns an enormous range into small, readable numbers.
Disclaimer: This calculator is for general information only. It evaluates expressions as entered; always check the angle mode, parentheses, and order of operations before relying on a result for school, work, or financial decisions.