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AHL 1.9—Log laws - IB Questionbank | RevisionDojo

Practice AHL 1.9—Log laws with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $\ln a=p, \ln b=q$ . Write the following expressions in terms of $p$ and $q$

$\ln a^{3} b$

[2]

$\ln \frac{\sqrt{a}}{b}$

[2]

Question 2

HLPaper 1

Find the value of the integer $k$ in each of the following cases:

$\log 3 k=\log 3+\log 5$

[1]

$\log 21=\log 3+\log k$

[1]

$\ln 21=\ln 3+\ln k$

[1]

Question 3

HLPaper 2

Let $\log _{b} 3=x$ and $\log _{b} 16=y$

Find an expression for $\log _{b} 9$ in terms of $x$ .

[2]

Find an expression for $\log _{b} 4$ in terms of $y$ .

[2]

Find an expression for $\log _{b} 48$ in terms of $x$ and $y$ .

[2]

Question 4

HLPaper 1

Let $p=\log x, q=\log y$ and $r=\log z$ .

Write $\log \frac{x}{y^{2} \sqrt{z}}$ in terms of $p, q$ and $r$ .

[3]

Question 5

HLPaper 1

Given that $\log _{a} 8=3$

Find the value of $\log _{a} 64$

[2]

Find the value of $a$ .

[2]

Find the exact value of $\log _{a^{2}} 8$ .

[3]

Question 6

HLPaper 1

Find the following integers using logarithm laws.

$\ln e^{5}$

[1]

$e^{\ln 5}$

[1]

$\log _{3} 3^{5}$

[1]

$3^{\log _{3} 5}$

[1]

$\log 10^{5}$

[1]

$10^{\log 5}$

[1]

$\log _{a} a^{5}$

[1]

$a^{\log _{a} 5}$

[1]

Question 7

HLPaper 1

Let $\log x=a, \log y=b$ and $\log z=c$ .

Write $\log \frac{x^{2} \sqrt{y}}{z^{3}}$ in terms of $a, b$ and c.

[3]

Question 8

HLPaper 1

Given that $p=\log _{a} 5, q=\log _{a} 2$ . Write the following expressions in terms of $p$ and/or $q$

$\log _{a} 10$

[2]

$\log _{a} 8$

[2]

$\log _{a} 2.5$

[2]

Question 9

HLPaper 1

Let $r=\log 2$ and $s=\log 12$ . Write down the following expressions in terms of $r$ and $s$ .

$\log 24$

[3]

$\log 3$

[3]

$\log 72$

[3]

Question 10

HLPaper 1

Let $\log _{c} 3=p, \log _{c} 5=q$ . Find an expression in terms of $p$ and $q$

$\log _{c} 15$

[2]

$\log _{c} 25$

[2]

Find the value of $d$ if $\log _{d} 6=\frac{1}{2}$

[2]

AHL 1.9—Log laws Questionbank