Log Calculator – Solve Logarithms Online Easily

Intro

The Log Calculator online is a free tool that helps you quickly calculate logarithms of any number to any base. A logarithm is the inverse of exponentiation and is widely used in mathematics, physics, computer science, finance, and engineering.

For example, the logarithm answers the question: log⁡b(x)=ymeansby=x\log_b(x) = y \quad \text{means} \quad b^y = xlogb​(x)=ymeansby=x

Manually solving logarithms can be difficult, especially with non-standard bases like log base 2 or log base 10. The Log Calculator online makes it fast and simple—you just enter the number and base, and it gives you the result instantly.

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Formula / Working

The calculator uses the logarithm definition and properties:

1. Basic Definition:

log⁡b(x)=y⇔by=x\log_b(x) = y \quad \Leftrightarrow \quad b^y = xlogb​(x)=y⇔by=x

2. Change of Base Formula:

log⁡b(x)=ln⁡(x)ln⁡(b)=log⁡10(x)log⁡10(b)\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}logb​(x)=ln(b)ln(x)​=log10​(b)log10​(x)​

3. Common Logarithms:

• Log base 10 (common log): log⁡(x)\log(x)log(x)
• Log base e (natural log): ln⁡(x)\ln(x)ln(x)
• Log base 2 (binary log): log⁡2(x)\log_2(x)log2​(x)

The calculator applies these formulas to find logarithms accurately for any positive number xxx and base b>0,b≠1b > 0, b \neq 1b>0,b=1.

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Step-by-Step Usage

Using the Log Calculator online is easy:

1. Enter the number (x) you want the log of.
2. Enter the base (b). Common bases are 2, 10, and e.
3. Click Calculate.
4. Instantly get the logarithm result.

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Examples

Example 1: Common Logarithm

log⁡10(1000)\log_{10}(1000)log10​(1000)

Since 103=100010^3 = 1000103=1000,
✅ Result = 3

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Example 2: Natural Logarithm

ln⁡(e2)\ln(e^2)ln(e2)

Since base eee is natural,
✅ Result = 2

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Example 3: Binary Logarithm

log⁡2(32)\log_2(32)log2​(32)

Since 25=322^5 = 3225=32,
✅ Result = 5

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Example 4: Non-standard Base

log⁡5(125)\log_5(125)log5​(125)

Since 53=1255^3 = 12553=125,
✅ Result = 3

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FAQs

Q1. What is a Log Calculator?
It’s an online tool that calculates logarithms of numbers with any base, including 2, 10, and e.

Q2. Can I calculate logs of decimals and fractions?
Yes, the calculator works with integers, decimals, and fractions.

Q3. Is this tool useful for students?
Absolutely—it’s great for algebra, calculus, statistics, and computer science.

Q4. What’s the difference between log and ln?

• log(x) usually means log base 10.
• ln(x) means log base e (natural logarithm).

Q5. Is the Log Calculator free to use?
Yes, it’s 100% free, fast, and works on any device.