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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:

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

[1]

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

[1]

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

[1]

Question 3

HLPaper 2

Consider the properties of logarithms, specifically the product and power rules.

Using the identity $\log_a(xy) = \log_a x + \log_a y$ , calculate $\log_2(32 \cdot 4)$ .

[4]

If $\log_a x = 3$ and $\log_a y = 2$ , find the value of $\log_a(x^2y^3)$ .

[3]

Question 4

HLPaper 1

Solve the equation $\ln 27 = 3 \ln x - 15$ . Give your answer in the form $p e^q$ , where $p$ and $q$ are positive integers.

[3]

Question 5

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 6

HLPaper 2

The force, $F$ newtons, between two magnets a distance $d$ metres apart is modelled by the equation $F = k d^n$ . The following measurements were taken:

$\begin{array}{|c|c|} \hline d & F \\ \hline 1 & 0.15 \\ 2 & 0.0375 \\ 3 & 0.0167 \\ 4 & 0.00938 \\ \hline \end{array}$

Linearize the relationship between $F$ and $d$ using the given model $F = k d^n$ .

[2]

Use linear regression to find the values of $k$ and $n$ , giving your answer to two significant figures.

[2]

Question 7

HLPaper 2

The curve $C$ is defined by the equation $xy - \ln y = 1$, $y > 0$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

[4]

Determine the equation of the tangent to $C$ at the point $\left( \frac{2}{\mathrm{e}}, \mathrm{e} \right)$.

[3]

Question 8

HLPaper 2

A computer scientist is analyzing the efficiency of algorithms and uses logarithmic functions to express their performance. In this question, give all answers as integers.

State the value of $\log_5 5$ .

[1]

Calculate $\log_5 5 + \log_5 25$ .

[2]

Find $\log_5 125 - \log_5 5$ .

[2]

Question 9

HLPaper 2

Consider the expression $x^{0.75} \cdot y^{1.25}$ .

Write the expression $\log(x^{0.75} \cdot y^{1.25})$ in terms of $\log(x)$ and $\log(y)$ .

[2]

If $x = 16$ and $y = 81$ , evaluate $x^{0.75} \cdot y^{1.25}$ .

[2]

Question 10

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]

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:

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

[1]

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

[1]

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

[1]

Question 3

HLPaper 2

Consider the properties of logarithms, specifically the product and power rules.

Using the identity $\log_a(xy) = \log_a x + \log_a y$ , calculate $\log_2(32 \cdot 4)$ .

[4]

If $\log_a x = 3$ and $\log_a y = 2$ , find the value of $\log_a(x^2y^3)$ .

[3]

Question 4

HLPaper 1

Solve the equation $\ln 27 = 3 \ln x - 15$ . Give your answer in the form $p e^q$ , where $p$ and $q$ are positive integers.

[3]

Question 5

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 6

HLPaper 2

The force, $F$ newtons, between two magnets a distance $d$ metres apart is modelled by the equation $F = k d^n$ . The following measurements were taken:

$\begin{array}{|c|c|} \hline d & F \\ \hline 1 & 0.15 \\ 2 & 0.0375 \\ 3 & 0.0167 \\ 4 & 0.00938 \\ \hline \end{array}$

Linearize the relationship between $F$ and $d$ using the given model $F = k d^n$ .

[2]

Use linear regression to find the values of $k$ and $n$ , giving your answer to two significant figures.

[2]

Question 7

HLPaper 2

The curve $C$ is defined by the equation $xy - \ln y = 1$, $y > 0$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

[4]

Determine the equation of the tangent to $C$ at the point $\left( \frac{2}{\mathrm{e}}, \mathrm{e} \right)$.

[3]

Question 8

HLPaper 2

A computer scientist is analyzing the efficiency of algorithms and uses logarithmic functions to express their performance. In this question, give all answers as integers.

State the value of $\log_5 5$ .

[1]

Calculate $\log_5 5 + \log_5 25$ .

[2]

Find $\log_5 125 - \log_5 5$ .

[2]

Question 9

HLPaper 2

Consider the expression $x^{0.75} \cdot y^{1.25}$ .

Write the expression $\log(x^{0.75} \cdot y^{1.25})$ in terms of $\log(x)$ and $\log(y)$ .

[2]

If $x = 16$ and $y = 81$ , evaluate $x^{0.75} \cdot y^{1.25}$ .

[2]

Question 10

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]

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:

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

[1]

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

[1]

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

[1]

Question 3

HLPaper 2

Consider the properties of logarithms, specifically the product and power rules.

Using the identity $\log_a(xy) = \log_a x + \log_a y$ , calculate $\log_2(32 \cdot 4)$ .

[4]

If $\log_a x = 3$ and $\log_a y = 2$ , find the value of $\log_a(x^2y^3)$ .

[3]

Question 4

HLPaper 1

Solve the equation $\ln 27 = 3 \ln x - 15$ . Give your answer in the form $p e^q$ , where $p$ and $q$ are positive integers.

[3]

Question 5

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 6

HLPaper 2

The force, $F$ newtons, between two magnets a distance $d$ metres apart is modelled by the equation $F = k d^n$ . The following measurements were taken:

$\begin{array}{|c|c|} \hline d & F \\ \hline 1 & 0.15 \\ 2 & 0.0375 \\ 3 & 0.0167 \\ 4 & 0.00938 \\ \hline \end{array}$

Linearize the relationship between $F$ and $d$ using the given model $F = k d^n$ .

[2]

Use linear regression to find the values of $k$ and $n$ , giving your answer to two significant figures.

[2]

Question 7

HLPaper 2

The curve $C$ is defined by the equation $xy - \ln y = 1$, $y > 0$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

[4]

Determine the equation of the tangent to $C$ at the point $\left( \frac{2}{\mathrm{e}}, \mathrm{e} \right)$.

[3]

Question 8

HLPaper 2

A computer scientist is analyzing the efficiency of algorithms and uses logarithmic functions to express their performance. In this question, give all answers as integers.

State the value of $\log_5 5$ .

[1]

Calculate $\log_5 5 + \log_5 25$ .

[2]

Find $\log_5 125 - \log_5 5$ .

[2]

Question 9

HLPaper 2

Consider the expression $x^{0.75} \cdot y^{1.25}$ .

Write the expression $\log(x^{0.75} \cdot y^{1.25})$ in terms of $\log(x)$ and $\log(y)$ .

[2]

If $x = 16$ and $y = 81$ , evaluate $x^{0.75} \cdot y^{1.25}$ .

[2]

Question 10

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]

AHL 1.9—Log laws Questionbank