Common and Natural Logarithms

The definition of a logarithm works with any positive base except , but in practice two bases account for nearly every logarithm you will meet. Base 10 matches our decimal number system, which is why it measures earthquakes, sound levels and acidity. Base matches the way things grow continuously, which is why it runs through calculus, finance and physics. The first is the common logarithm; the second is the natural logarithm.

What Is the Common Logarithm?

The common logarithm is the logarithm with base . It is written without a base:

What does log10 mean? It asks how many factors of make up a number. For exact powers of ten the answer is a whole number:

For everything in between, the log is a decimal. says that sits between and , closer to than to on a multiplicative scale. That is why the common log measures orders of magnitude: the whole-number part tells you roughly how many digits a number has.

What Is the Natural Logarithm?

The natural logarithm is the logarithm with base , written :

It is the inverse of the natural exponential function . Some values to recognise:

The natural log of 2, , is worth remembering. It appears in every doubling-time and half-life formula, because solving gives .

Natural Log vs Common Log

Both are defined only for positive inputs, both equal at , and both are negative between and . The acidity scale in the pH calculator is a common logarithm in action: . So is the Richter scale for earthquakes, where each whole step means ten times the shaking.

Natural Log Rules With e

Because and are inverses, four rules follow directly from the definition:

The product, quotient and power rules — and the rest — hold for exactly as for any base, and are covered in the logarithm rules lesson.

How to Get Rid of a Natural Log

To undo a natural log in an equation, raise to both sides. The and the cancel, leaving the argument:

In practice you skip the middle step and convert straight to exponential form. The same move undoes a common log, with in place of : gives .

Going the other way — to get rid of — take the natural log of both sides. becomes .

Converting Between log and ln

Any logarithm can be rewritten in terms of another base. For these two:

For example, . This is a special case of the change of base formula, which converts between any two bases.

Can a Natural Log Be Negative?

The output can; the input cannot.

• For : .
• For : .
• For : . For instance .
• For : is undefined. There is no natural log of or of a negative number, because is positive for every real .

As shrinks toward , falls without bound toward . The y-axis is a vertical asymptote of the graph.

Worked Example

Solve , giving the answer to four decimal places.

Step 1 — undo the natural log by converting to exponential form:

Step 2 — evaluate the power:

Step 3 — solve the linear equation:

Step 4 — check that the argument is positive:

So .

Common Mistakes to Avoid

• Treating and as interchangeable. , not . Only .
• Dividing by to cancel it. does not give . is a function, not a number; undo it by exponentiating.
• Writing . . is undefined.
• Forgetting to check the argument. After solving, substitute back and make sure what sits inside every is positive.

Practice Problems

Work each one before opening the answer.

Problem 1. Evaluate and .

Problem 2. Simplify and .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Without a calculator, decide whether is positive or negative, and whether exists.

Everything here rests on the few rules that let logarithms be split up and recombined. The logarithm rules lesson sets out the product, quotient and power rules and shows how to expand and condense log expressions.