Log Calculator – Step by Step with Examples

Log Calculator to solve log equations instantly with step-by-step. Supports natural logs, base 10, and custom bases. Ideal for students and professionals

log

A logarithm calculator is a quick and accurate online tool that helps you solve logarithmic expressions without manual effort. It is designed to calculate logarithms with base 10, natural logs (ln), and any custom base.

This calculator not only gives you the final result but also shows step-by-step solutions so you can understand the calculation process better.

🔹 What is a Log Calculator

In simple terms, a log Calculator answers the question:

“To what power must a base be raised, to get a certain number?”

Mathematical definition:

log⁡b(x)=yif and only ifby=x\log_b(x) = y \quad \text{if and only if} \quad b^y = xlogb​(x)=yif and only ifby=x

Where:

  • bbb = base (greater than 0, not equal to 1)
  • xxx = the number (must be positive)
  • yyy = exponent (result of the log)

✅ Examples:

  • log⁡2(8)=3\log_2(8) = 3log2​(8)=3 because 23=82^3 = 823=8
  • log⁡10(1000)=3\log_{10}(1000) = 3log10​(1000)=3 because 103=100010^3 = 1000103=1000
  • ln⁡(e5)=5\ln(e^5) = 5ln(e5)=5 because natural log uses base eee

🔹 How to Use the Log Calculator?

Using the calculator is simple:

  1. Enter the base (e.g., 2, 10, or eee).
  2. Enter the value you want the log of.
  3. Click Calculate.
  4. View the result and detailed steps instantly.
Log Calculator.
Log Calculator
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🔹 Step-by-Step Examples

Example 1: Natural Logarithm (ln⁡\lnln)

Find ln⁡(56)\ln(56)ln(56).

Steps:

  1. Base = eee (≈ 2.718).
  2. Formula: ln⁡(56)\ln(56)ln(56).
  3. Calculation:

ln⁡(56)≈4.025\ln(56) ≈ 4.025ln(56)≈4.025

✅ Final Answer: log⁡e(56)≈4.025\log_e(56) ≈ 4.025loge​(56)≈4.025.

Example 2: Logarithm with Base 2

Find log⁡2(56)\log_2(56)log2​(56).

Steps:

  1. Apply the change of base formula:

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

  1. Compute:
    ln⁡(56)≈4.025\ln(56) ≈ 4.025ln(56)≈4.025
    ln⁡(2)≈0.693\ln(2) ≈ 0.693ln(2)≈0.693
  2. Divide:
    4.0250.693≈5.797\frac{4.025}{0.693} ≈ 5.7970.6934.025​≈5.797

✅ Final Answer: log⁡2(56)≈5.797\log_2(56) ≈ 5.797log2​(56)≈5.797.

Example 3: Common Logarithm (log⁡10\log_{10}log10​)

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

Steps:

  1. Recognize 103=100010^3 = 1000103=1000.
  2. Therefore:

log⁡10(1000)=3\log_{10}(1000) = 3log10​(1000)=3

✅ Final Answer: log⁡10(1000)=3\log_{10}(1000) = 3log10​(1000)=3.

🔹Benefits of Using the Log Calculator

  • ✅ Saves time and effort.
  • ✅ Explains results step by step.
  • ✅ Works with natural logs, base-10 logs, and custom bases.
  • ✅ Useful for students, teachers, engineers, and researchers.

FAQs About Logarithm Calculator

Q1. What is the default base of a log?

  • By default, log means base 10. Natural log (ln⁡\lnln) means base eee.

Q2. Can I use decimal or fractional bases?

  • Yes, the calculator accepts any base greater than 0 and not equal to 1.

Q3. Why can’t I calculate log of a negative number?

  • Because logarithms are only defined for positive numbers.

Q4. What is ln in the calculator?

  • “ln” means logarithm with base eee (Euler’s number ≈ 2.718).

Q5. Where are logarithms used in real life?

  • In finance (compound interest), physics (sound intensity, earthquakes), biology (growth models), and computer science (algorithm complexity).

Disclaimer

This Logarithm Calculator is provided for educational purposes. While it gives accurate results for most values, always verify critical calculations in academic or professional work. Logs of negative numbers or base = 1 are undefined.

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