How many times do you multiply 10 by itself to get 1000? Three times. That is what log(1000) = 3 means. What is the largest number that divides 48, 60 and 72 without a remainder? Twelve. And the area of a triangle with base 12 and height 8? Exactly 48 square units.
Three problems, three calculators, zero guesswork. Each one shows the formula, the steps and the result.
Logarithm Calculator - four bases, one tool
Logarithms reverse exponentiation. Instead of asking "what is 10 to the power of 3?", you ask "what power of 10 gives 1000?" The answer is the logarithm.
The calculator supports four logarithm types:
| Type | Notation | Base | Where you see it |
|---|---|---|---|
| Common | log(x) | 10 | pH scale, decibels, Richter scale |
| Natural | ln(x) | e (2.718...) | Calculus, compound interest, physics |
| Binary | log2(x) | 2 | Computer science, bits, algorithm complexity |
| Custom | log_a(x) | any a > 0 | Specific math problems |
A quick reference for values worth remembering:
| Input | log10 | ln | log2 |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.693 | 1 |
| 10 | 1 | 2.303 | 3.322 |
| 100 | 2 | 4.605 | 6.644 |
| 1000 | 3 | 6.908 | 9.966 |
The change-of-base formula connects all types: log_a(x) = log(x) / log(a). You only need one type to calculate any other. The calculator handles conversions automatically.
Where does this matter outside a textbook? Sound intensity in decibels uses log10. The pH scale for acidity is -log10 of hydrogen ion concentration. Binary search in computer science runs in O(log2 n) time. And compound interest doubling time uses the natural log: years = ln(2) / ln(1 + rate).
GCD & LCM Calculator - from fractions to scheduling
Greatest Common Divisor and Least Common Multiple are two sides of the same coin. GCD finds the largest shared factor. LCM finds the smallest shared multiple.
Enter 48, 60, 72 and the calculator returns GCD = 12 and LCM = 720. The Euclidean algorithm does the heavy lifting - repeatedly dividing and taking remainders until it reaches zero.
The relationship between GCD and LCM is elegant: GCD(a,b) x LCM(a,b) = a x b. Knowing one gives you the other instantly.
| Problem | Uses | Example |
|---|---|---|
| Simplifying fractions | GCD | 48/72 = (48/12)/(72/12) = 4/6 = 2/3 |
| Finding common denominator | LCM | 1/48 + 1/60 needs LCD = LCM(48,60) = 240 |
| Scheduling | LCM | Bus A every 48 min, Bus B every 60 min - both at stop together every 240 min |
| Tiling a floor | GCD | Room 48x60 cm - largest square tile is 12x12 cm |
| Gear ratios | GCD | 48-tooth and 72-tooth gears - ratio 4:6 simplified by GCD=12 |
Two numbers with GCD = 1 are called coprime. They share no common factors. Any two consecutive integers are always coprime. This property is fundamental in cryptography - RSA key generation requires checking that two numbers are coprime.
Area Calculator - seven shapes, one formula table
Pick a shape, enter dimensions, get the area. No need to remember whether the trapezoid formula divides by 2 or multiplies by it.
Triangle with base 12 and height 8: area = (12 x 8) / 2 = 48 square units. The calculator shows the formula used and the unit.
All seven supported shapes at a glance:
| Shape | Formula | Quick example |
|---|---|---|
| Square | a^2 | Side 5 -> area 25 |
| Rectangle | a x b | 8 x 3 -> area 24 |
| Triangle | (base x height) / 2 | 12 x 8 / 2 -> area 48 |
| Circle | pi x r^2 | r=7 -> area 153.94 |
| Trapezoid | (a + b) x h / 2 | (6+10) x 4 / 2 -> area 32 |
| Parallelogram | base x height | 12 x 5 -> area 60 |
| Rhombus | (d1 x d2) / 2 | 10 x 8 / 2 -> area 40 |
The height of a triangle is the perpendicular distance from base to opposite vertex - not the length of a side. This is the most common mistake. For a trapezoid, both parallel sides (bases) are needed plus the perpendicular height between them.
For irregular shapes, break them into simpler components. An L-shaped room? Two rectangles. A house floor plan? Rectangles plus triangles for the roof section. Calculate each piece, add them up.
Tools featured in this article
Logarithm Calculator - Calculate log base 10, natural log (ln), binary log (log2) or any custom base. Step-by-step formula with verification.
GCD & LCM Calculator - Find the Greatest Common Divisor and Least Common Multiple of two or more numbers. Euclidean algorithm shown step by step.
Area Calculator - Calculate the area of 7 geometric shapes: square, rectangle, triangle, circle, trapezoid, parallelogram, rhombus.
More Math tools
- Percentage Calculator
- Pythagorean Theorem Calculator
- Fraction Calculator
- Average Calculator
- Standard Deviation Calculator
- Proportion Calculator
- Powers & Roots Calculator
Check for a specific number
- Log of 10 - common logarithm
- Log of 100 - powers of 10
- Log of 1000 - scientific notation