LogarithmsIB MYP Maths Extended: Revision notes

A logarithm answers the question: what power of the base gives this number? $\log_a b=c\iff a^c=b\quad(a>0,\ a\neq1,\ b>0).$ For example $2^5=32$, so $\log_2 32=5$. Also $10^3=1000$, so $\log_{10}1000=3$. A logarithm is the inverse of exponentiation, just as division is the inverse of multiplication. The base is written as a small subscript. On a calculator, the $\log$ key uses base 10.

Ask 'what power of the base gives the number?' Write the number as a power of the base. $\log_5 25=2$ because $5^2=25$. $\log_2\frac18=-3$ because $2^{-3}=\frac18$. Two special values hold for every base: $\log_a1=0$ because $a^0=1$, and $\log_a a=1$ because $a^1=a$. You cannot take the logarithm of zero or of a negative number.

For positive $x$ and $y$, with the same base: $\log_a x+\log_a y=\log_a(xy),\qquad \log_a x-\log_a y=\log_a\!\left(\frac xy\right).$ Adding logarithms means multiplying the numbers; subtracting means dividing. Example: $\log_{10}4+\log_{10}25=\log_{10}100=2$. Example: $\log_3 54-\log_3 2=\log_3 27=3$.

A power inside a logarithm comes out as a multiplier: $\log_a(x^n)=n\log_a x.$ Example: $\log_{10}8=\log_{10}2^3=3\log_{10}2$. If $\log_{10}2=a$, then $\log_{10}8=3a$. Combine the laws: $2\log_5 10-\log_5 4=\log_5\frac{100}{4}=\log_5 25=2$. In the same way, if $\log_{10}2=a$ and $\log_{10}3=b$ then $\log_{10}\frac43=2a-b$.

An exponential equation has the unknown in the index, such as $2^x=7$. Take logarithms of both sides, then use the power law: $x\log 2=\log 7\ \Rightarrow\ x=\frac{\log7}{\log2}=2.81\ (3\text{ s.f.}).$ This is the same as $x=\log_2 7$. First isolate the power: $500\times3^t=20\,000$ becomes $3^t=40$, then $t=\frac{\log40}{\log3}=3.36$. You can also use technology. Check by substituting back: $2^{2.81}\approx7$.

Logarithms describe quantities that grow over huge ranges. Sound level is $L=10\log_{10}\left(\frac{I}{10^{-12}}\right)$ decibels. Each time the intensity is multiplied by 10, the level rises by 10 dB. Doubling the intensity adds $10\log_{10}2\approx3.01$ dB. Use the definition to undo a logarithm: if $\log_{10}x=11$ then $x=10^{11}$. Give units, round sensibly and say what the answer means in context. (Natural logarithms, $\ln$ and $e$, are enrichment and are not assessed.)