LOG10

Returns the base-10 logarithm of a number.

Syntax:

LOG10(number)

returns the logarithm to base 10 of number.

Example:

LOG10(5)

returns the base-10 logarithm of 5 (approximately 0.69897).

Application:

pH and Acidity

The log10 function, also known as the common logarithm, is a powerful tool for working with numbers that span many orders of magnitude. A classic application of LOG10 is its use in the pH scale, which measures the acidity or alkalinity of a substance.

The Concept:

The concentration of hydrogen ions ([H+]) in a solution determines its acidity. In a highly acidic solution, the concentration of H+ ions is very high, while in an alkaline solution, it is very low. These concentrations can range from 100 moles per liter (very acidic) to 10−14 moles per liter (very alkaline). Working with these small, negative exponents can be cumbersome.

The pH scale was developed to simplify this by using the log10 function. The formula for pH is:

pH=−log10[H+]

The negative sign is included so that the pH value is a positive number. When we take the log10 of the hydrogen ion concentration, we are essentially finding the power to which 10 must be raised to get that concentration.

Example: Common Household Liquids

Let's look at the hydrogen ion concentration and the corresponding pH for a few common liquids.

	Substance	Hydrogen Ion Concentration ([H+]) in Moles/Liter	Calculation	pH
	A	B	C	D
1	Stomach Acid	10-1	pH=−log10(10−1)=−(−1)=1	1
2	Lemon Juice	10-2	pH=−log10(10−2)=−(−2)=2	2
3	Black Coffee	10-5	pH=−log10(10−5)=−(−5)=5	5
4	Pure Water	10-7	pH=−log10(10−7)=−(−7)=7	7
5	Baking Soda	10-9	pH=−log10(10−9)=−(−9)=9	9
6	Household Ammonia	10-11	pH=−log10(10−11)=−(−11)=11	11

Why log10 is so Useful Here:

Simplifies Data: Instead of dealing with numbers like 0.00000001, we get simple, manageable integers like 8. This makes the scale much easier to understand and use.
Represents Proportional Changes: Each whole number step on the pH scale represents a tenfold change in hydrogen ion concentration. For example, a substance with a pH of 3 is ten times more acidic than a substance with a pH of 4. This relationship is a direct consequence of the base-10 nature of the logarithm.
Visualizes a Broad Range: The log10 function compresses a vast range of numbers (100 to 10−14) into a small, easy-to-read scale (0 to 14). This allows scientists and students to easily compare substances with vastly different acidities on a single chart.