Topics
Logarithm algebra
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Definition and notation of logarithms, including logab=x⇔ax=b, common and natural logs log and ln, log laws for products, quotients and powers, evaluating logs exactly or with technology, and using logarithms to solve exponential equations.
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Definition of the logarithm
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.
Evaluating logs algebraically
AHL AI 1.9
Some logarithms can be evaluated by hand using the fact that
logab=x⇒ax=b
For example, we can find log279 by solving the equation
27x=9⇒33x=32⇒x=32
Log base 10
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.
Natural logarithm
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.
Evaluating logs using technology
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
Sum and difference of logs
AHL AI 1.9
The sum of logs with the same base is the log of the products:
logax+logay=loga(xy)📖
We have a similar rule for the difference of logs:
logax−logay=loga(yx)📖
Log power rule
AHL AI 1.9
loga(xm)=mlogax📖
Using logs to solve exponential equations
SL Core 1.5
Logarithms can be used to solve exponential equations:
ax=b⇔x=logab.
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