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Logarithmic scale: pH scale

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pH Scale: A Logarithmic Scale Explained

This lesson explains the pH scale as an application of logarithms. You will see the formula that links pH to hydrogen ion concentration, understand why the scale is logarithmic rather than linear, and work through examples calculating pH and concentration in both directions.

What makes the pH scale logarithmic?

The pH scale is one of the most common real-world uses of logarithms. It measures how acidic or basic a solution is, based on the concentration of hydrogen ions, \([\)H\(^+]\), dissolved in it. Hydrogen ion concentrations can range from about \(1\) mol/L in a very strong acid down to \(0.00000000000001\) mol/L in a very strong base. Writing and comparing numbers across that many decimal places is awkward, so chemists compress that huge range into a small, easy-to-read scale (usually from 0 to 14) using a logarithm.

If you have not yet studied what a logarithm is, it is worth reviewing that idea first, since the entire pH scale is really just one logarithmic formula applied to chemistry.

The pH of a solution is defined as:

\(\)pH\( = -\log_{10}[\)H\(^+]\)

Here, \([\)H\(^+]\) is the hydrogen ion concentration measured in moles per liter (mol/L). Because concentration values are small decimals, the logarithm is negative, so the formula flips the sign to give a positive, easy-to-read pH value.

The formula can also be rearranged into exponential form to solve for concentration when the pH is known:

\([\)H\(^+] = 10^{-\)pH\(}\)

Because the pH formula uses \(\log_{10}\), moving one whole unit on the pH scale corresponds to multiplying or dividing the hydrogen ion concentration by exactly 10. A solution with pH 4 has ten times more hydrogen ions than a solution with pH 5, and one hundred times more than a solution with pH 6. This is exactly what "logarithmic scale" means: equal steps on the scale represent equal ratios (not equal differences) in the underlying quantity.

The graph below shows how hydrogen ion concentration falls off as pH increases from 0 to 14. Notice how quickly the curve drops near pH 0, this is the signature shape of an exponential relationship, which is the inverse of the logarithmic pH formula.

Graph of hydrogen ion concentration equals 10 to the power of negative x, for pH values x from 0 to 14 Plot of y = 10**(-x) for x in [0, 14] 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 pH value Hydrogen ion concentration (mol/L) pH 0, highly acidic pH 7, neutral pH 13, highly basic
Hydrogen ion concentration versus pH, showing the exponential decay described by \([\)H\(^+] = 10^{-\)pH\(}\).

Most common substances fall somewhere between pH 0 and pH 14, with pH 7 marking a neutral solution such as pure water.

0 7 14 Acidic Neutral Basic e.g. lemon juice pure water e.g. bleach

Suppose a solution has a hydrogen ion concentration of \([\)H\(^+] = 1 \times 10^{-3}\) mol/L. Find its pH.

\(\)pH\( = -\log_{10}(1 \times 10^{-3})\)

\(\)pH\( = -(-3) = 3\)

The solution has a pH of 3, which is fairly acidic. If you want more practice with logarithm calculations like this one without reaching for a calculator, see evaluating logs without a calculator.

Suppose a solution has a pH of 5.5. Find its hydrogen ion concentration.

\([\)H\(^+] = 10^{-5.5}\)

\([\)H\(^+] \approx 3.16 \times 10^{-6}\ \)mol/L\(\)

This calculation is really just the pH formula solved as a logarithmic equation for \([\)H\(^+]\). Since these problems are so common in chemistry courses, it can help to practice the underlying algebra separately, as covered in solving logarithmic equations.

The 0 to 14 range is a convenient guideline, not a hard boundary. Because the pH formula is just \(-\log_{10}[\)H\(^+]\), a solution can technically have a pH below 0 if its hydrogen ion concentration is greater than 1 mol/L, which happens in some very concentrated industrial acids. This is why you may see the term "negative pH scale": it is not a separate scale, just the same logarithmic formula extended below zero for exceptionally strong acids.

The pH scale exists because logarithms let scientists represent an enormous range of hydrogen ion concentrations using small, manageable numbers. Once you are comfortable moving between logarithmic and exponential form, calculating pH in either direction becomes a straightforward substitution into \(\)pH\( = -\log_{10}[\)H\(^+]\) or its inverse, \([\)H\(^+] = 10^{-\)pH\(}\).

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