Logarithm Studio
Solve for any base (10, 2, e) and prove the Laws of Logarithms interactively.
Base-N Logarithm Solver
Calculate logarithms for any base using the Change of Base formula.
log_b(x) = ln(x) / ln(b)
Unlocking the Power of Logs
For centuries before computers, Logarithms were the calculator of the world. They transformed impossible multiplication tasks into simple addition. Today, they are still the secret sauce behind sound scales, earthquake measurements, and information theory.
The Inverse
Division undoes Multiplication. Subtraction undoes Addition. **Logarithms undo Exponents**. If 2³ = 8, then log₂(8) = 3. They find the "How Many".
Compressing Scale
Logs are great at squashing huge numbers into workable sizes. A "6-figure salary" is a logarithmic concept (log₁₀(100,000) = 5, log₁₀(1,000,000) = 6).
Binary Logs
in Computer Science, **Base 2** is king. The binary log (lg n) tells us the minimum bits needed to store a number, or the steps needed in a binary search.
The 3 Laws of Logarithms
Master these three rules, and you can dismantle almost any exponential equation. (Try the **"Log Laws"** tab above to prove them yourself!)
- 1. Product Rule: log(xy) = log(x) + log(y)
"The log of a product is the sum of the logs." - 2. Quotient Rule: log(x/y) = log(x) - log(y)
"The log of a quotient is the difference of the logs." - 3. Power Rule: log(x^n) = n · log(x)
"You can drag the exponent down to the front."
Frequently Asked Questions
What is a Logarithm?
A logarithm is simply an exponent in disguise. It asks: 'How many of this number (base) do we multiply to get that number?'. For example, log₂(8) = 3 because 2 × 2 × 2 = 8.
What is the Common Log vs Natural Log?
Common Log uses Base 10 and is written as just log(x). It's used in science (pH, Richter scale). Natural Log uses Base e (~2.718) and is written as ln(x). It's used in growth calculations.
Why is log(1) always 0?
Because any base raised to the power of 0 equals 1. So, the question 'Base to what power is 1?' is always answered by 0.
Can you take the log of a negative number?
In the real number system, no. You cannot multiply positive numbers (multiples of a positive base) to equal a negative result. Complex numbers are needed for that.
What is the Change of Base Formula?
It allows you to calculate strange bases using a standard calculator: log_b(x) = log(x) / log(b) or ln(x) / ln(b).
What is the Product Rule?
log(a · b) = log(a) + log(b). This turns complicated multiplication problems into simpler addition problems.
What is the Quotient Rule?
log(a / b) = log(a) - log(b). It turns division problems into subtraction.
What is the Power Rule?
log(a^n) = n · log(a). This is powerful for bringing exponents 'down to earth' to solve algebraic equations.
How do logs relate to the Richter Scale?
The Richter scale is logarithmic. A magnitude 7 earthquake is 10 times stronger than a magnitude 6, and 100 times stronger than a magnitude 5.
Why is log(0) undefined?
Because you can never multiply a base enough times to reach zero. You approach negative infinity, but never touch zero.
What is a Binary Logarithm?
It is log base 2 (lg n). It is fundamental in Computer Science, measuring bits of information or the depth of binary trees.
How do I calculate fractional logs?
Fractional logs represent roots. For example, log₄(2) = 0.5 because √4 = 2 (or 4^0.5 = 2).