Logarithms and Solving Equations

The logarithm of x to the base a is written "log_a x" and is defined as follows.

log_a x = b if and only if a^b = x
where a > 0 (a 1), and x >0 .

Note that the base must be positive and different from 1, and the expression that you are taking the logarithm of must also be positive.

Some logarithms can be evaluated easily. For example, log₄ 64 = 3 since 4³ = 64.

Special Bases

If the log base is 10, then the log is called the common logarithm and we write "log" for log₁₀.

If the log base is the number "e", then the logarithm is called the natural logarithm and we write "ln" for log_e. The "ln" key on a scientific calculator gives values for the natural logarithm. For example, ln(32) = 3.466 to three decimal places, as determined using a calculator.

Properties of Logarithms

Here, a > 0 and a 1.

1. log_a1 = 0
2. log_a a = 1
3. log_a(a^x) = x
4. a^(log_ax) = x, if x > 0
5. log_ax + log_a y = log_a(xy) if x, y > 0
6. log_ax - log_a y = log_a (x/y) if x, y > 0
7. log_a (x^r) = r log_a x if x > 0

Please note: these properties work "both" ways.

Some examples of these properties.

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