Scientific Calculator: Functions Beyond the Four Operations
Most people use calculators for basic arithmetic. The scientific functions come out infrequently enough that it's worth having a reference for what each one does.
Exponents and Roots
xⁿ or x^y: x raised to the power y. 2^10 = 1024.
√x: square root. √144 = 12.
ⁿ√x: nth root. ∛27 = 3 (cube root of 27).
eˣ: natural exponential. e ≈ 2.71828. e^1 = 2.71828.
Logarithms
log (log₁₀): common logarithm (base 10). log(1000) = 3 because 10³ = 1000.
ln: natural logarithm (base e). ln(e) = 1.
Relationship: ln(x) = log(x) / log(e) ≈ log(x) / 0.4343
Practical use: "How many digits does this number have?" digits ≈ floor(log₁₀(n)) + 1. The number 1,000,000 has 7 digits; log₁₀(1,000,000) = 6; floor(6) + 1 = 7. ✓
Trigonometry
sin, cos, tan: ratios in a right triangle. sin(θ) = opposite/hypotenuse. cos(θ) = adjacent/hypotenuse. tan(θ) = opposite/adjacent.
Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹, also written arcsin, arccos, arctan): given a ratio, returns the angle.
Important: most calculators default to degrees mode. Scientific calculations in physics and calculus typically use radians. Check your mode.
Factorial (n!)
n! = n × (n-1) × (n-2) × ... × 2 × 1
5! = 5 × 4 × 3 × 2 × 1 = 120
Used in probability (permutations and combinations) and series expansions.
Large factorials get enormous fast: 20! ≈ 2.43 × 10¹⁸.
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Frequently Asked Questions
Why does my calculator give a different answer for sin(90) depending on the mode?
Your calculator is switching between degrees and radians mode. In degrees mode, sin(90°) = 1, but in radians mode, sin(90 radians) ≈ 0.89. For trigonometry in physics and calculus, you'll typically need radians, but basic geometry uses degrees. Always check which mode your calculator is in before calculating.
How do I find how many digits are in a really large number?
Use the common logarithm (log₁₀) formula: digits ≈ floor(log₁₀(n)) + 1. For example, the number 1,000,000 has log₁₀(1,000,000) = 6, so floor(6) + 1 = 7 digits. This works for any positive number without having to count manually.
What's the difference between log and ln on a scientific calculator?
Log (or log₁₀) uses base 10, while ln uses base e (approximately 2.71828). They're related by the formula ln(x) = log(x) / 0.4343. Use log for practical problems like counting digits or engineering applications, and use ln in calculus, physics, and natural growth/decay problems.
Why does my calculator say 20! is so huge?
Factorials multiply very quickly. 20! = 20 × 19 × 18 × ... × 2 × 1 ≈ 2.43 × 10¹⁸, which is 2.43 quintillion. Factorials are used in counting problems (permutations and combinations), but the numbers grow so fast that even modest factorials become astronomical—this is why calculators show them in scientific notation.
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