Logarithm algebra

SL Core 1.5

Logarithms are a mathematical tool for asking "what power of a given base gives a specific value". We write this as

logab=x⇔ax=b.

Here, a is called the base, and it must be positive and not equal to 1. b must also be positive. The value of x, however, can be any real number.

SL Core 1.5

In science and mathematics, it is so common to use log10 that we can simply write the shorthand log to indicate log10.

For example, log(0.001)=−3 since 10−3=0.001.

SL Core 1.5

Another special logarithm is the one in base e. We call it the natural logarithm due to the fundamental importance of e across mathematics.

loge is the same as ln

For example, ln(e3)=3.

SL Core 1.5

If a and b are not powers of the same base, the log cannot be easily computed by hand. But we can use a calculator to evaluate them approximately.

log35≈1.46

SL Core 1.5

Logarithms can be used to solve exponential equations:

ax=b⇔x=logab.

Exercises