Common Logarithms (Log Base 10)

A common logarithm is simply a logarithm written in base 10. Instead of writing \(\log_{10}x\) every time, mathematicians agree that when no base is shown, the base is understood to be 10. So \(\log x\) always means \(\log_{10}x\).

If you haven't yet seen the general definition of a logarithm, it helps to start with what a logarithm is before working through this lesson, since common logarithms are just a special, very useful case.

Base 10 matches our decimal number system, so common logarithms show up constantly in real measurements: the pH scale in chemistry, the Richter scale for earthquakes, and the decibel scale for sound all use \(\log_{10}\). That's why this particular base earned its own shorthand notation.

Two ways of writing the exact same thing:

Every logarithm question is really an exponent question in disguise. Because \(\log x\) has an unwritten base of 10, the statement \(\log x = y\) always means \(10^y = x\). If you're not yet comfortable flipping between these two forms, it's worth practicing how to convert a logarithm to exponential form first.

For example:

• \(\log 100 = 2\) because \(10^2 = 100\)
• \(\log 1000 = 3\) because \(10^3 = 1000\)
• \(\log 1 = 0\) because \(10^0 = 1\)
• \(\log 0.1 = -1\) because \(10^{-1} = 0.1\)

Whenever the input is a power of 10, you can evaluate a common log by inspection: just count how many times 10 is multiplied by itself. \(\log 10{,}000 = 4\) because \(10{,}000 = 10^4\). For inputs that aren't neat powers of 10, you often need to rewrite the expression using log rules or estimate between two whole numbers. For a deeper set of strategies, see evaluating logarithms without a calculator.

Nearly every scientific or graphing calculator has a button labeled "LOG". Pressing it computes the common logarithm directly. For example, entering \(\log 50\) gives approximately \(1.699\), since \(10^{1.699} \approx 50\). This is different from the "LN" button, which computes the natural logarithm (base \(e\)) instead, so common and natural logarithms should never be mixed up when reading a calculator screen.

Plotting \(y = \log x\) shows the shape shared by every base 10 logarithm: the curve rises slowly, passes through \((1, 0)\) because \(\log 1 = 0\), and never touches the y-axis, since \(\log x\) is undefined for \(x \le 0\).

Common logarithms follow the same product, quotient, and power rules as logarithms in any other base. To combine or expand common log expressions, the same techniques used for the general logarithm apply, such as the rules explained in the lesson on the product rule of logarithms. Once an equation is written with a single common log on each side, you solve it by converting back to exponential form. For a full walkthrough of that process, check out how to solve logarithmic equations.

Solve \(\log x = 3\) for \(x\).

Since \(\log x\) means \(\log_{10}x\), rewrite the equation in exponential form: \(10^3 = x\), so \(x = 1000\).

A common logarithm is a base 10 logarithm, written without a subscript as \(\log x\). It answers "10 to what power gives x?" and appears throughout science and engineering because our number system is base 10. Mastering how to move between logarithmic and exponential form makes evaluating and solving with common logs straightforward.

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