Practice IB Mathematics Analysis and Approaches (AA) Topic SL 1.7—laws of Exponents and Logs with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 1.7—laws of Exponents and Logs and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 1

Consider the infinite series $S = lo g_{a} (1) + lo g_{a} (r) + lo g_{a} (r^{2}) + lo g_{a} (r^{3}) + \dots,$ where $a > 1$ and $0 < r < 1$ .

Show that the $n$ -th partial sum can be written as $S_{n} = lo g_{a} (r^{0 + 1 + 2 + \dots + (n - 1)}) .$

[3]

Deduce that $S_{n} = \frac{n ( n - 1 )}{2} lo g_{a} (r) .$ Hence, explain why the infinite series diverges to $- \infty$ .

[3]

Question 2

SLPaper 2

Consider a geometric sequence with $u_{1} = \frac{18}{5}$ , $r > 0$ , and $u_2 u_3 u_4=\frac{729}{64}.$

Find $r$ .

[4]

Find the least value of $n$ such that $u_{n} < 1$ .

[4]

Question 3

SL & HLPaper 1

Given that $lo g_{5} K = m$ and $lo g_{25} (5 K) = n$ ,

Show that $m = 2 n - 1$ .

[4]

Question 4

SLPaper 1

A geometric sequence $(u_{n})$ has $u_{1} = \frac{9}{2}$ , $r > 0$ , and $u_2u_4u_6=\frac{1}{27}.$

Determine $r$ .

[4]

Hence find the least integer $n$ such that $u_{n} < \frac{1}{54}$ . Leave your answer in terms of logarithms.

[4]

Question 5

SLPaper 2

A geometric sequence $(u_{n})$ has $u_{1} = 3$ and $r > 0$ . Given that $u_{2} u_{5} u_{8} = \frac{27}{64}$ ,

Find the exact value of $r$ .

[4]

Hence, using your value of $r$ , find the smallest integer $n$ such that $u_{n} < \frac{1}{100}$ (exact form with logs acceptable).

[4]

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