Laws of Logarithms II
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Solve the following, giving exact answers where possible:
(a)\(3^x = 20\)
(b)\(5^{2x-1} = 30\)
(c)\(e^{3x} = 12\)
(d)\(2e^x + 3 = 11\)
(a)
Answer
\(x = \dfrac{\log 20}{\log 3} pprox 2.73\)
(b)
Take log
\((2x-1)\log 5=\log 30\Rightarrow 2x-1=\dfrac{\log 30}{\log 5}\)
Answer
\(x=\dfrac{1}{2}\!\left(\dfrac{\log 30}{\log 5}+1
ight)pprox 1.49\)
(c)
Answer
\(x=\dfrac{\ln 12}{3}\)
Question 2
Solve simultaneously: \(\log_2(x+y)=3\) and \(\log_2(x-y)=1\).
Convert
\(x+y=8\) and \(x-y=2\)
Question 3
Solve \(\log_3(x^2-4) - \log_3(x+2) = 1\).
Factorise and combine
\(\log_3\dfrac{x^2-4}{x+2}=\log_3(x-2)=1\Rightarrow x-2=3\)
Question 4
Find the exact value of \(\log_4 8 + \log_8 4\).
Change of base to 2
\(\log_4 8 = frac{3}{2};\quad \log_8 4 = frac{2}{3}\)
Answer
\(\dfrac{3}{2}+\dfrac{2}{3}=\dfrac{13}{6}\)
Question 5
The population of a city is modelled by \(P = 200\,000 imes 1.03^t\) where \(t\) is time in years.
(a)Find the population after 10 years.
(b)Find the number of years until the population doubles.
(a)
Answer
\(200\,000 imes1.03^{10}pprox268\,783\)
(b)
Solve
\(1.03^t=2\Rightarrow t=\dfrac{\ln 2}{\ln 1.03}\)
Answer
\(tpprox23.4\) years
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