Content
Logarithm algebra
Definition and evaluation of logarithms, properties of logs, e and the natural log, change of base rule
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Definition and evaluation of logarithms, properties of logs, e and the natural log, change of base rule
Want a deeper conceptual understanding? Try our interactive lesson!
No exercises available for this concept.
Logarithms are a mathematical tool for asking "what power of a given base gives a specific value". We write this as
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.
Some logarithms can be evaluated by hand using the fact that
For example, we can find log279 by solving the equation
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.
Another special logarithm is the one in base e. We call it the natural logarithm due to the fundamental importance of e across mathematics.
For example, ln(e3)=3.
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.
The sum of logs with the same base is the log of the products:
We have a similar rule for the difference of logs:
We can change the base of a logarithm using the law
for any choice of positive b=1.
Logarithms can be used to solve exponential equations: