Mathski
Worksheet
Introduction to Logarithms
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
Convert the following to logarithmic form:
(a)\(2^5 = 32\)
(b)\(10^3 = 1000\)
(c)\(3^{-2} = frac{1}{9}\)
(a)
Answer
\(\log_2 32 = 5\)
(b)
Answer
\(\log_{10} 1000 = 3\)
(c)
Answer
\(\log_3 frac{1}{9} = -2\)
Question 2
Evaluate without a calculator:
(a)\(\log_2 16\)
(b)\(\log_5 125\)
(c)\(\log_3 frac{1}{27}\)
(d)\(\log_{10} 0.001\)
(e)\(\log_4 8\)
(f)\(\log_9 3\)
(a)
Answer
\(4\)
(b)
Answer
\(3\)
(c)
Answer
\(-3\)
(d)
Answer
\(-3\)
(e)
Write as power of 2
\(4^x = 8 \Rightarrow 2^{2x} = 2^3 \Rightarrow x = frac{3}{2}\)
Answer
\( frac{3}{2}\)
(f)
Write as power of 3
\(9^x = 3 \Rightarrow 3^{2x}=3^1 \Rightarrow x= frac{1}{2}\)
Answer
\( frac{1}{2}\)
Question 3
Solve for \(x\):
(a)\(\log_3 x = 4\)
(b)\(\log_x 64 = 3\)
(c)\(\log_2(x-1) = 5\)
(a)
Answer
\(x = 81\)
(b)
Convert
\(x^3 = 64\)
Answer
\(x = 4\)
(c)
Convert
\(x-1 = 2^5 = 32\)
Answer
\(x = 33\)
Question 4
Sketch the graph of \(y = \log_2 x\), stating the domain, range, and any asymptotes or intercepts.
Key features
Domain: \(x>0\). Range: \(\mathbb{R}\). Vertical asymptote: \(x=0\). Passes through \((1,0)\) and \((2,1)\).
Answer
Increasing curve, x-intercept at \((1,0)\), passes through \((2,1)\) and \((4,2)\).
Question 5
Using the change of base formula, evaluate \(\log_3 20\) to 3 significant figures.
Change of base
\(\log_3 20 = \dfrac{\ln 20}{\ln 3} = \dfrac{2.996...}{1.099...}\)
Answer
\(2.73\)
Generated by Mathski  ·  mathski.io  ·  IB Mathematics AA Shadow Worksheets