ganak.appLog Calculator
Free log calculator for any base. Compute logarithms, anti-logs, find the base, and change of base with step-by-step solutions and verification. Includes common log values table, logarithm rules, and real-world applications. Supports log₂, ln, log₁₀, and custom bases.
Equivalent Forms
The same relationship expressed differently
Step-by-Step Solution
See how the answer is calculated — free, no paywall
log_b(x) = y means b^y = x
We need y such that 10^y = 100
log₁₀(x) = log(x)
log₁₀(100)
= 2
10^2 should equal 100
10^2 = 100
= Verified ✓
Log Values for Base 10
Quick reference table of common logarithm values
Common Logarithm Values
Reference table across log₂, ln, and log₁₀
| x | log₂(x) | ln(x) | log₁₀(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0.6931 | 0.301 |
| e | 1.4427 | 1 | 0.4343 |
| 5 | 2.3219 | 1.6094 | 0.699 |
| 10 | 3.3219 | 2.3026 | 1 |
| 50 | 5.6439 | 3.912 | 1.699 |
| 100 | 6.6439 | 4.6052 | 2 |
| 500 | 8.9658 | 6.2146 | 2.699 |
| 1000 | 9.9658 | 6.9078 | 3 |
How to Calculate Logarithms
Understanding base, argument, and the logarithm formula
A logarithm answers the question: "What power must we raise the base to in order to get a given number?" In the expression logb(x) = y, we are saying that by = x. The base b must be positive and not equal to 1.
Change of Base Formula
logb(x) = ln(x) / ln(b) = log(x) / log(b)
This calculator supports four modes: computing a logarithm (log), computing an anti-logarithm (inverse log), finding an unknown base, and converting between different bases. Results include step-by-step solutions with verification.
Three common logarithm bases are: base 10 (common log, written as "log"), base e (natural log, written as "ln", where e ≈ 2.71828), and base 2 (binary log, used in computer science).
Logarithm Rules & Properties
The fundamental rules that govern how logarithms work
Product Rule
logb(xy) = logb(x) + logb(y)
Log of a product equals the sum of the logs
Quotient Rule
logb(x/y) = logb(x) − logb(y)
Log of a quotient equals the difference of the logs
Power Rule
logb(xn) = n · logb(x)
The exponent can be brought in front as a multiplier
Change of Base
logb(x) = logc(x) / logc(b)
Convert any logarithm base using a different base
Reciprocal Rule
logb(1/x) = −logb(x)
Log of a reciprocal is the negative of the log
Identity & Zero
logb(b) = 1 & logb(1) = 0
Log of the base is 1; log of 1 is always 0
Common Mistakes with Logarithms
Avoid these frequent errors when working with logs
log(a + b) ≠ log(a) + log(b)
The product rule applies to multiplication, not addition. log(a + b) cannot be simplified. Only log(a × b) = log(a) + log(b).
log(0) is undefined, not 0
No power of a positive base can equal zero. As x approaches 0 from the right, log(x) approaches −∞. Similarly, log of a negative number is undefined in real numbers.
Confusing log and ln notation
In most contexts, "log" means base 10, and "ln" means base e. But in some fields (pure math, CS), "log" may mean base e or base 2. Always check the convention.
Base must be positive and ≠ 1
The base of a logarithm must satisfy b > 0 and b ≠ 1. If b = 1, then 1 raised to any power is always 1, so log1(x) is undefined for x ≠ 1.
Real-World Applications of Logarithms
Where logarithms appear in science, engineering, and everyday life
pH Scale (Chemistry)
pH = −log₁₀[H⁺]. A pH of 3 means [H⁺] = 10⁻³ = 0.001 mol/L. Each pH unit represents a 10× change in acidity.
Richter Scale (Seismology)
Earthquake magnitude uses log₁₀. A magnitude 6 earthquake has 10× the amplitude of a magnitude 5, and 31.6× more energy.
Decibels (Sound & Signal)
dB = 10 · log₁₀(P₁/P₀). A 10 dB increase means 10× the power. Human hearing spans ~120 dB, a trillion-fold power range.
Computer Science (Big-O)
Binary search runs in O(log₂ n) time. To search 1 billion items, you need at most log₂(10⁹) ≈ 30 comparisons.
Frequently Asked Questions
Common questions about logarithms and how to calculate them