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SL 1.5—Exponents and Logarithms - IB Questionbank | RevisionDojo

Practice SL 1.5—Exponents and Logarithms 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 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 2

HLPaper 2

The function $f(x)=\ln\left(\frac{1}{x-2}\right)$ is defined for $x>2, \ x \in \mathbb{R}$ .

Find an expression for $f^{-1}(x)$ . You are not required to state a domain.

[4]

Solve $f(x)=f^{-1}(x)$ .

[1]

Question 3

SLPaper 1

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$, which have a magnitude of at least $M$. For a particular region the equation is

$$ \log_{10} N = a - M \text{, for some } a \in \mathbb{R} $$

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$.

The equation for this region can also be written as $N = \frac{b}{10^M}$.

The expected length of time, in years, between earthquakes with a magnitude of at least $M$ is $\frac{1}{N}$.

Within this region the most severe earthquake recorded had a magnitude of $7.2$.

Find the value of $a$.

[2]

Find the value of $b$.

[2]

Given $0 < M < 8$, find the range for $N$.

[2]

Find the expected length of time between this earthquake and the next earthquake of at least this magnitude. Give your answer to the nearest year.

[2]

Question 4

SLPaper 1

The amount, in milligrams, of a medicinal drug in the body $t$ hours after it was injected is given by $D(t) = 23(0.85)^t, \ t \geq 0$ . Before this injection, the amount of the drug in the body was zero.

Write down the initial dose of the drug.

[1]

Write down the percentage of the drug that leaves the body each hour.

[2]

Calculate the amount of the drug remaining in the body 10 hours after the injection.

[2]

Question 5

HLPaper 1

It is believed that two variables, $m$ and $p$, are related. Experimental values of $m$ and $p$ are obtained. A graph of $\ln m$ against $p$ shows a straight line passing through $(2.1, 7.3)$ and $(5.6, 2.4)$.

Hence, find:

the equation of the straight line, giving your answer in the form $\ln m = ap + b$, where $a, b \in \mathbb{R}$.

[3]

a formula for $m$ in terms of $p$.

[1]

the value of $m$ when $p = 0$.

[1]

Question 6

SLPaper 2

A car depreciates in value exponentially. The initial value of the car is $20,000, and it depreciates at a rate of 15% per year.

Calculate the value of the car after 5 years.

[2]

Determine the time it takes for the car's value to halve.

[2]

Question 7

SLPaper 2

Give your answers to parts 2, 3 and 4 to the nearest whole number.

Harinder has 14 000 US Dollars (USD) to invest for a period of five years. He has two options of how to invest the money.

Option A: Invest the full amount, in USD, in a fixed deposit account in an American bank.

The account pays a nominal annual interest rate of r% , compounded yearly, for the five years. The bank manager says that this will give Harinder a return of 17 500 USD.

Option B: Invest the full amount, in Indian Rupees (INR), in a fixed deposit account in an Indian bank. The money must be converted from USD to INR before it is invested.

The exchange rate is 1 USD = 66.91 INR.

The account in the Indian bank pays a nominal annual interest rate of 5.2 % compounded monthly.

Calculate the value of r.

[3]

Calculate 14 000 USD in INR.

[2]

Calculate the amount of this investment, in INR, in this account after five years.

[3]

Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.

[3]

Question 8

HLPaper 2

Consider the equation $x^5 - 3x^4 + mx^3 + nx^2 + px + q = 0$ , where $m, n, p, q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as $\log_2 a$ , $\log_2 b$ and $\log_2 c$ .

The equation also has two imaginary roots, one of which is $di$ where $d \in \mathbb{R}$ .

The values $a, b, c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8d^2$ , find the other two real roots.

[9]

Question 9

HLPaper 1

Solve the equation $4^x + 2^{x+2} = 3$ .

[6]

Question 10

SLPaper 1

The $pH$ of a solution is given by the formula $pH = -\log_{10} C$ where $C$ is the hydrogen ion concentration in a solution, measured in moles per litre ($\text{mol L}^{-1}$).

Find the $pH$ value for a solution in which the hydrogen ion concentration is $5.2 \times 10^{-8}$.

[2]

Write an expression for $C$ in terms of $pH$.

[2]

Find the hydrogen ion concentration in a solution with $pH = 4.2$. Give your answer in the form $a \times 10^k$ where $1 \leq a < 10$ and $k$ is an integer.

[2]

Paper

Level

Question 1

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 2

HLPaper 2

The function $f(x)=\ln\left(\frac{1}{x-2}\right)$ is defined for $x>2, \ x \in \mathbb{R}$ .

Find an expression for $f^{-1}(x)$ . You are not required to state a domain.

[4]

Solve $f(x)=f^{-1}(x)$ .

[1]

Question 3

SLPaper 1

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$, which have a magnitude of at least $M$. For a particular region the equation is

$$ \log_{10} N = a - M \text{, for some } a \in \mathbb{R} $$

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$.

The equation for this region can also be written as $N = \frac{b}{10^M}$.

The expected length of time, in years, between earthquakes with a magnitude of at least $M$ is $\frac{1}{N}$.

Within this region the most severe earthquake recorded had a magnitude of $7.2$.

Find the value of $a$.

[2]

Find the value of $b$.

[2]

Given $0 < M < 8$, find the range for $N$.

[2]

Find the expected length of time between this earthquake and the next earthquake of at least this magnitude. Give your answer to the nearest year.

[2]

Question 4

SLPaper 1

Write down the initial dose of the drug.

[1]

Write down the percentage of the drug that leaves the body each hour.

[2]

Calculate the amount of the drug remaining in the body 10 hours after the injection.

[2]

Question 5

HLPaper 1

Hence, find:

the equation of the straight line, giving your answer in the form $\ln m = ap + b$, where $a, b \in \mathbb{R}$.

[3]

a formula for $m$ in terms of $p$.

[1]

the value of $m$ when $p = 0$.

[1]

Question 6

SLPaper 2

A car depreciates in value exponentially. The initial value of the car is $20,000, and it depreciates at a rate of 15% per year.

Calculate the value of the car after 5 years.

[2]

Determine the time it takes for the car's value to halve.

[2]

Question 7

SLPaper 2

Give your answers to parts 2, 3 and 4 to the nearest whole number.

Harinder has 14 000 US Dollars (USD) to invest for a period of five years. He has two options of how to invest the money.

Option A: Invest the full amount, in USD, in a fixed deposit account in an American bank.

The account pays a nominal annual interest rate of r% , compounded yearly, for the five years. The bank manager says that this will give Harinder a return of 17 500 USD.

Option B: Invest the full amount, in Indian Rupees (INR), in a fixed deposit account in an Indian bank. The money must be converted from USD to INR before it is invested.

The exchange rate is 1 USD = 66.91 INR.

The account in the Indian bank pays a nominal annual interest rate of 5.2 % compounded monthly.

Calculate the value of r.

[3]

Calculate 14 000 USD in INR.

[2]

Calculate the amount of this investment, in INR, in this account after five years.

[3]

Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.

[3]

Question 8

HLPaper 2

Consider the equation $x^5 - 3x^4 + mx^3 + nx^2 + px + q = 0$ , where $m, n, p, q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as $\log_2 a$ , $\log_2 b$ and $\log_2 c$ .

The equation also has two imaginary roots, one of which is $di$ where $d \in \mathbb{R}$ .

The values $a, b, c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8d^2$ , find the other two real roots.

[9]

Question 9

HLPaper 1

Solve the equation $4^x + 2^{x+2} = 3$ .

[6]

Question 10

SLPaper 1

The $pH$ of a solution is given by the formula $pH = -\log_{10} C$ where $C$ is the hydrogen ion concentration in a solution, measured in moles per litre ($\text{mol L}^{-1}$).

Find the $pH$ value for a solution in which the hydrogen ion concentration is $5.2 \times 10^{-8}$.

[2]

Write an expression for $C$ in terms of $pH$.

[2]

Find the hydrogen ion concentration in a solution with $pH = 4.2$. Give your answer in the form $a \times 10^k$ where $1 \leq a < 10$ and $k$ is an integer.

[2]

Paper

Level

Question 1

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 2

HLPaper 2

The function $f(x)=\ln\left(\frac{1}{x-2}\right)$ is defined for $x>2, \ x \in \mathbb{R}$ .

Find an expression for $f^{-1}(x)$ . You are not required to state a domain.

[4]

Solve $f(x)=f^{-1}(x)$ .

[1]

Question 3

SLPaper 1

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$, which have a magnitude of at least $M$. For a particular region the equation is

$$ \log_{10} N = a - M \text{, for some } a \in \mathbb{R} $$

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$.

The equation for this region can also be written as $N = \frac{b}{10^M}$.

The expected length of time, in years, between earthquakes with a magnitude of at least $M$ is $\frac{1}{N}$.

Within this region the most severe earthquake recorded had a magnitude of $7.2$.

Find the value of $a$.

[2]

Find the value of $b$.

[2]

Given $0 < M < 8$, find the range for $N$.

[2]

Find the expected length of time between this earthquake and the next earthquake of at least this magnitude. Give your answer to the nearest year.

[2]

Question 4

SLPaper 1

Write down the initial dose of the drug.

[1]

Write down the percentage of the drug that leaves the body each hour.

[2]

Calculate the amount of the drug remaining in the body 10 hours after the injection.

[2]

Question 5

HLPaper 1

Hence, find:

the equation of the straight line, giving your answer in the form $\ln m = ap + b$, where $a, b \in \mathbb{R}$.

[3]

a formula for $m$ in terms of $p$.

[1]

the value of $m$ when $p = 0$.

[1]

Question 6

SLPaper 2

A car depreciates in value exponentially. The initial value of the car is $20,000, and it depreciates at a rate of 15% per year.

Calculate the value of the car after 5 years.

[2]

Determine the time it takes for the car's value to halve.

[2]

Question 7

SLPaper 2

Give your answers to parts 2, 3 and 4 to the nearest whole number.

Harinder has 14 000 US Dollars (USD) to invest for a period of five years. He has two options of how to invest the money.

Option A: Invest the full amount, in USD, in a fixed deposit account in an American bank.

The account pays a nominal annual interest rate of r% , compounded yearly, for the five years. The bank manager says that this will give Harinder a return of 17 500 USD.

Option B: Invest the full amount, in Indian Rupees (INR), in a fixed deposit account in an Indian bank. The money must be converted from USD to INR before it is invested.

The exchange rate is 1 USD = 66.91 INR.

The account in the Indian bank pays a nominal annual interest rate of 5.2 % compounded monthly.

Calculate the value of r.

[3]

Calculate 14 000 USD in INR.

[2]

Calculate the amount of this investment, in INR, in this account after five years.

[3]

Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.

[3]

Question 8

HLPaper 2

Consider the equation $x^5 - 3x^4 + mx^3 + nx^2 + px + q = 0$ , where $m, n, p, q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as $\log_2 a$ , $\log_2 b$ and $\log_2 c$ .

The equation also has two imaginary roots, one of which is $di$ where $d \in \mathbb{R}$ .

The values $a, b, c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8d^2$ , find the other two real roots.

[9]

Question 9

HLPaper 1

Solve the equation $4^x + 2^{x+2} = 3$ .

[6]

Question 10

SLPaper 1

The $pH$ of a solution is given by the formula $pH = -\log_{10} C$ where $C$ is the hydrogen ion concentration in a solution, measured in moles per litre ($\text{mol L}^{-1}$).

Find the $pH$ value for a solution in which the hydrogen ion concentration is $5.2 \times 10^{-8}$.

[2]

Write an expression for $C$ in terms of $pH$.

[2]

Find the hydrogen ion concentration in a solution with $pH = 4.2$. Give your answer in the form $a \times 10^k$ where $1 \leq a < 10$ and $k$ is an integer.

[2]

SL 1.5—Exponents and Logarithms Questionbank