Logarithms

Logarithms, or "logs", are a simple way of expressing numbers in terms of a single base. Common logs are done with base ten, but some logs ("natural" logs) are done with the constant "e" as their base. The log of any number is the power to which the base must be raised to give that number. In other words:

For example, log(10) is 1 and log(100) is 2 (because 10² = 100). Logs can easily be found for either base on your calculator. Usually there are two different buttons, one saying "log", which is base ten, and one saying "ln", which is a natural log, base e. It is always assumed, unless otherwise stated, that "log" means log₁₀.

The opposite of a log is the antilog, which means to raise the base to that number. Antilogs "undo" logarithms.

Logs are read aloud as "log", "natural log", "ln", or "log base whatever". To read log₃4, you would simply say "log, base three, of four".

Logs are commonly used in chemistry. The most prominent example is the pH scale. The pH of a solution is the -log([H⁺]), where square brackets mean concentration.

There are two major kinds of equations that you will have to solve using logs. In one kind, you will know the log of a number and have to find the number by taking anti-logs, which means raising the base to a power. The other kind gives you the variable in the exponent, and you have to take logs to isolate it. Solving these kinds of problems depends on knowing another property of logs: if the log of a number with an exponent is taken, then the log of that number is multiplied by whatever was in the exponent.

Sometimes, you may be required to convert between bases. Using some simple algebra, a formula can be derived for changing bases:

So, use these properties to solve the problems below:

Logs also have some unusual properties which allow you to combine them more easily. The log of one number times the log of another number is equal to the log of the first plus the second number, as long as the logs have the same base. Similarly, the log of one number divided by the log of another number is equal to the log of the first number minus the second (again, as long as the logs have the same base).

That's it! These basic concepts of logs can be applied in many different situations in chemistry, as you may see later on. Use the practice problems below to check out your understanding of this reading.

Try It Out:

Problem 1:

1. Solve the following equation:
log(x^2) - log 10 - 3 = 0

Solution is available.

Problem 2:

2. Simplify the following expression to three significant figures:
log₅9 + log₂3 + log₂6

Solution is available.

Problem 3:

3. Solve the following problem.
7 = ln5x + ln(7x-2x)

Solution is available.

[Numbers and their Properties] [Numbers in Science] [Ratios and Proportions]
[Units, Dimensions, and Conversions] [Percents] [Simple Statistics] [Logarithms]

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