What is a Logarithm?

What is a Logarithm?

Let’s start with a familiar problem from exponents:

$2^3 = 8$

We know this means:

• Multiply 2 by itself 3 times: $2 × 2 × 2 = 8$
• The number 3 is called the exponent
• The number 2 is called the base

Now, what if we want to solve this problem:

$2^x = 8$

We need to find x. In other words: “What power must we raise 2 to, in order to get 8?”

This is exactly what logarithms help us solve! We write it as:

$\log_2(8) = 3$

Three Common Types of Logarithms

1. Base 10 Logarithm (written as $\log$ or $\log_{10}$)

   • Most common in everyday calculations
   • Example: $\log_{10}(100) = 2$ because $10^2 = 100$

2. Natural Logarithm (written as $\ln$)

   • Uses base $e$ (approximately 2.718)
   • Very important in calculus
   • Example: $\ln(e) = 1$ because $e^1 = e$

3. Base 2 Logarithm (written as $\log_2$)

   • Used in computer science
   • Example: $\log_2(8) = 3$ because $2^3 = 8$

Basic Rules

1. Product Rule: When multiplying, add the logs $\log(a × b) = \log(a) + \log(b)$

2. Quotient Rule: When dividing, subtract the logs $\log(a ÷ b) = \log(a) - \log(b)$

3. Power Rule: Powers become multiplication $\log(a^n) = n × \log(a)$

Real-World Applications

📏 Measuring Earthquakes

The Richter scale uses logarithms (base 10):

• A magnitude 6 earthquake is 10 times stronger than magnitude 5
• A magnitude 7 earthquake is 100 times stronger than magnitude 5

地震マグニチュードの比較

🔊 Sound Levels (Decibels)

• Every +10 decibels means the sound is 10 times louder
• Normal conversation: ~60 dB
• Rock concert: ~110 dB (about 100,000 times louder!)

Want to Practice?

Try our logarithm calculator to solve logarithm problems step by step.