Laws of Logarithms I
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Write the following as a single logarithm:
(a)\(\log 3 + \log 5\)
(b)\(\log_2 12 - \log_2 3\)
(c)\(3\log 2 + \log 5\)
(d)\(2\log_3 x - \log_3 y + frac{1}{2}\log_3 z\)
(c)
Power rule first
\(\log 8 + \log 5 = \log 40\)
(d)
Answer
\(\log_3\dfrac{x^2\sqrt{z}}{y}\)
Question 2
Expand as sums/differences of logs:
(a)\(\log\!\left(\dfrac{x^3}{y^2}
ight)\)
(b)\(\ln\!\left(x^2\sqrt{y+1}
ight)\)
(c)\(\log_2\!\left(\dfrac{4\sqrt{x}}{y^3}
ight)\)
(a)
Answer
\(3\log x - 2\log y\)
(b)
Answer
\(2\ln x + frac{1}{2}\ln(y+1)\)
(c)
Expand
\(\log_2 4 + \log_2 x^{1/2} - \log_2 y^3 = 2 + frac{1}{2}\log_2 x - 3\log_2 y\)
Answer
\(2 + frac{1}{2}\log_2 x - 3\log_2 y\)
Question 3
Solve the following equations (give exact answers):
(a)\(\log_2 x + \log_2(x-2) = 3\)
(b)\(\log(3x+1) - \log(x-1) = 1\)
(c)\(2\ln x = \ln(3x-2)\)
(a)
Combine
\(\log_2[x(x-2)]=3\Rightarrow x(x-2)=8\Rightarrow x^2-2x-8=0\Rightarrow(x-4)(x+2)=0\)
(b)
Combine
\(\log frac{3x+1}{x-1}=1\Rightarrow frac{3x+1}{x-1}=10\Rightarrow 3x+1=10x-10\)
(c)
Power rule
\(\ln x^2=\ln(3x-2)\Rightarrow x^2=3x-2\Rightarrow x^2-3x+2=0\Rightarrow(x-1)(x-2)=0\)
Question 4
Given that \(\log_a 2 = p\) and \(\log_a 3 = q\), express the following in terms of \(p\) and \(q\):
(a)\(\log_a 6\)
(b)\(\log_a 18\)
(c)\(\log_a frac{4}{3}\)
(b)
Write \(18=2 imes3^2\)
\(p+2q\)
Question 5
Solve \(2^x = 7\), giving your answer to 3 significant figures.
Take log of both sides
\(x\log 2 = \log 7\Rightarrow x=\dfrac{\log 7}{\log 2}\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets