Logarithm Calculator

What is a Logarithm?

Math can often feel like learning a new language. If you understand addition, you learn subtraction to undo it. If you understand multiplication, you learn division to undo it. So, what undoes an exponent?

The answer is the Logarithm. A logarithm answers the question: "To what power must we raise the base to get a specific number?"

For example, we know that 10² = 100.

The logarithm asks the reverse: "10 raised to what power equals 100?"

The 3 Common Types of Logs

While you can have a logarithm with any base number, three specific bases appear so frequently in science, engineering, and nature that they have their own names and buttons on calculators.

1. Common Logarithm (Base 10)

If you see "log x" written without a base number, it is assumed to be Base 10. This is the standard logarithm used in high school math and engineering.

It scales by powers of 10:

• log(10) = 1
• log(100) = 2
• log(1000) = 3

This is used for the Richter Scale (earthquakes), pH Scale (acidity), and Decibels (sound).

2. Natural Logarithm (Base e)

This is written as "ln" rather than "log". The base is Euler's Number (e ≈ 2.71828).

The Natural Log is the language of growth. It appears in biology (population growth), physics (radioactive decay), and finance (compound interest).

3. Binary Logarithm (Base 2)

Written as log₂, this is the language of Computer Science. Since computers process information in binary (0s and 1s), this logarithm calculates bits. For example, log₂(8) = 3 means it takes 3 bits to represent 8 distinct values.

The 3 Essential Rules of Logarithms

Just like algebra has rules for simplifying equations, logs have three "Laws" that allow you to break complex expressions into simpler parts. These are crucial for calculus.

Real World Application: The Richter Scale

Why do we use logs? We use logarithms when we need to measure things that vary across huge scales.

Consider Earthquakes. The energy difference between a small tremor and a city-destroying quake is massive. If we used a linear scale, a small quake might be "1" and a big quake might be "1,000,000,000." These numbers are hard to work with.

Instead, we use a Logarithmic Scale (Base 10):

• A magnitude 5.0 quake is 10 times stronger than a 4.0 quake.
• A magnitude 6.0 quake is 100 times stronger than a 4.0 quake (10 × 10).

How to Convert Between Exponential and Logarithmic Form

Students often struggle to switch between the two forms. Here is the trick: remember the "circular" motion.

Start at the base (b), go across the equal sign to the answer (y), and that equals the argument (x).
"Base to the power of the answer equals the number."

Limitations: The Domain of a Log

If you try to type "-5" into our calculator, you will get an error. Why?

You cannot take the logarithm of a negative number or zero. Think about it in reverse: Is there any power you can raise 10 to that equals 0? No. 10⁰ = 1, and negative exponents just create fractions.