OMPT drill — Functions

OMPT exponential functions practice: lesson and 10 problems

Growth and decay, solving a^x = b, and recognising when an exponential model applies. Pairs directly with logarithms — study them back to back. Attempt each problem before opening the solution; the notation only becomes automatic by writing it yourself.

Tested inOMPT-A OMPT-B OMPT-D OMPT-E OMPT-F OMPT-G

OMPT exponential functions: the lesson

What the OMPT tests in exponential functions

An exponential function $f (x) = b \cdot g^{x}$ describes anything that grows or shrinks by the same factor each step: interest, populations, radioactive decay, a cooling cup of coffee. The $b$ is the starting amount, what you have at $x = 0$ , since $g^{0} = 1$ . The $g$ is the growth factor per step: $g = 1.04$ means up 4% each period, $g = 0.85$ means down 15%. Growth when $g > 1$ , decay when $0 < g < 1$ . The chapter "Exponential functions and logarithms" sits on OMPT-A, B, D, E, F and G; only OMPT-C leaves it out.

Three sub-skills carry the topic. Solving $g^{x} = c$ by writing both sides with the same base, which is the heaviest and appears on every paper. Building a model from a story or from two data points, then evaluating it or reading it back into words. And handling time units: doubling times, half-lives, a yearly rate asked per month. The question forms are "solve for $x$ ", "how much after $t$ years", "what is the growth factor", and "after how many periods does it fall below". The topic is won or lost on the translation between percentage and factor, because the exam speaks in percentages and the mathematics runs on factors.

Notation and answer format the OMPT expects

A growth factor is typed as a decimal or fraction: 1.04, 0.85, $1/2$ . A percentage is typed as the number: 25 for "25%".
Solutions of $g^{x} = c$ are exact: $x = 4$ , $x = - 3/4$ , or a power such as $2^{1/3}$ when no whole exponent works. Decimals only when the question says "round to".
Money is given to the cent when asked, so $562.43$ , and a non-integer time stays in the exponent: after half a year the balance is $500 \cdot 1.0 4^{0.5}$ , not $500 \cdot 1.02$ .
A formula answer is typed as $N (t) = 200 \cdot 0. 9^{t}$ , with the variable in the exponent.
If $g^{x}$ is set equal to zero or a negative number, the answer is "no solution", not 0.

Worked example 1: isolate, then match bases

Solve $5 \cdot 2^{x} = 80$ .
Isolate the exponential first; divide both sides by 5: $2^{x} = 16$ .
Recognise the base: $16 = 2^{4}$ , so $2^{x} = 2^{4}$ and $x = 4$ .
Check: $5 \cdot 16 = 80$ .
The order matters. Dividing by the 5 first turns it into a one-line finish. The other trap is dividing 80 by 2 and answering $x = 40$ .

Worked example 2: from story to number

An account holds $€ 500$ and grows by 4% per year. How much is in it after 3 years, to the cent?
Translate the percentage: growth factor $g = 1.04$ . Model: $A (t) = 500 \cdot 1.0 4^{t}$ .
Evaluate at $t = 3$ : $1.0 4^{2} = 1.0816$ and $1.0816 \cdot 1.04 = 1.124864$ .
So $A (3) = 500 \cdot 1.124864 = 562.432$ , which is $€ 562.43$ .
Sanity check: three flat steps of $€ 20$ would give $€ 560$ . The exponential answer is a little above that, because each year earns on the previous growth. An exponential answer below the linear estimate means the factor is wrong.

The full lesson adds base matching with fractions, roots and two different bases, rebuilding the factor from two data points, decay stories and the total-loss trap, half-lives and doubling times solved without logarithms, changing the time unit, and an equation that is a quadratic in disguise. It ends with the edge cases the OMPT likes and the slips I mark most often.

How each OMPT variant tests exponential functions

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Testedroughly 4 to 6 of 52 questions, shared with logarithms
Official chapter: Exponential functions and logarithms, 10% of the paper.
Solve g to the power x equals c by matching bases, build a growth or decay model from a percentage, evaluate it, and read a formula back into words. Short numeric answers.
OMPT-B
64 questions
Testedroughly 4 to 6 of 64 questions, shared with logarithms
Official chapter: Exponential functions and logarithms, 10% of the paper.
Same shapes as OMPT-A plus equations with two different bases, a quadratic in 2 to the power x, and exponential functions that reappear in the differentiation and integration chapters.
OMPT-C
30 questions
Not tested
OMPT-C has no exponential chapter; percentage change appears only as arithmetic in the Numbers chapter. Study exponents-and-roots and linear equations instead.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, equations and inequalities, 30% of the paper.
An exponential model opens a long question that then asks for a derivative, a limit or a tangent. Expect a parameter in the base or the exponent and a "for which values" part.
OMPT-E
40 questions
Testedroughly 3 to 4 of 40 questions, shared with logarithms
Official chapter: Exponential functions and logarithms, 5% of the paper.
Growth stories with a percentage per period: population, interest, depreciation. Evaluate after t periods, find the factor from two data points, and say when a level is reached.
OMPT-F
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, 15% of the paper.
Business framing: compound interest, depreciation, a sales model with a growth factor, then a derivative of the same function. Multi-part with the model built in part (a).
OMPT-G
26 questions
Testedroughly 2 to 3 of 26 questions
Official chapter: Functions, graphs and equations, 45% of the paper.
Exponentials arrive through chemistry and physics: radioactive decay and half-life, a cooling law, and geometric sequences from the sequences chapter. Give the number with its unit.

OMPT exponential functions practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

A bacteria culture starts at 800 cells and doubles every 5 hours. How many cells are there after 15 hours?

Show the worked solution

Answer: 6400

In 15 hours the culture doubles $15/5 = 3$ times, so the count is $800 \cdot 2^{3} = 800 \cdot 8 = 6400$ cells. I count doubling periods first, then apply the power — never multiply by 2 once per hour.

Problem 2Multiple choice

Which of these functions describes exponential decay?

A $y = 6 \cdot (0.85)^{x}$
B $y = 0.5 \cdot (1.2)^{x}$
C $y = 6 \cdot x^{- 2}$
D $y = 85 - 6 x$

Show the worked solution

Answer: A — $y = 6 \cdot (0.85)^{x}$

Exponential decay needs the form $y = a \cdot g^{x}$ with a growth factor $0 < g < 1$ . Only the first option qualifies: $g = 0.85$ . The second grows ( $g = 1.2$ — a small starting value does not make it decay), the third is a power function of $x$ , and the fourth is linear.

Problem 3Numeric answer

Solve for $n$ : $2^{n + 3} = 32$ .

Show the worked solution

Answer: 2

I write 32 as a power of 2: $32 = 2^{5}$ . Then $2^{n + 3} = 2^{5}$ forces $n + 3 = 5$ , so $n = 2$ . Check: $2^{5} = 32$ .

Problem 4Multiple choice

A town of 2500 people grows by $4%$ per year. Which formula gives the population after $t$ years?

A $N (t) = 2500 \cdot 1.0 4^{t}$
B $N (t) = 2500 \cdot 0.0 4^{t}$
C $N (t) = 2500 + 1.04 t$
D $N (t) = 2500 \cdot 4^{t}$

Show the worked solution

Answer: A — $N (t) = 2500 \cdot 1.0 4^{t}$

Growing by $4%$ means each year the population is multiplied by $1 + 0.04 = 1.04$ . After $t$ years that multiplication has happened $t$ times: $N (t) = 2500 \cdot 1.0 4^{t}$ . The growth factor always carries the "1 plus" — the 1 is the population you already had.

Problem 5Spot the error

A sample of 3 grams doubles 4 times. A student computes the final mass. Step 1: final mass $= (3 \cdot 2)^{4}$ . Step 2: $= 6^{4}$ . Step 3: $= 1296$ grams. Which step contains the error?

Show the worked solution

Answer: Step 1

The exponent belongs to the growth factor alone, not to the starting amount. Step 1 should read $3 \cdot 2^{4}$ , which is $3 \cdot 16 = 48$ grams. Steps 2 and 3 correctly evaluate the wrong expression. Walking through the doublings by hand confirms it: $3 \to 6 \to 12 \to 24 \to 48$ .

Problem 6Numeric answer

A medicine has a half-life of 6 hours. A patient takes 160 mg. How many milligrams remain after 18 hours?

Show the worked solution

Answer: 20

Eighteen hours contain $18/6 = 3$ half-life periods, so the amount is halved three times: $160 \cdot (\frac{1}{2})^{3} = 160/8 = 20$ mg. Step by step: $160 \to 80 \to 40 \to 20$ .

Problem 7Numeric answer

Solve for $x$ : $2 7^{x} = 9^{x + 1}$ .

Show the worked solution

Answer: 2

Both sides are powers of 3: $27 = 3^{3}$ and $9 = 3^{2}$ . So $3^{3 x} = 3^{2 (x + 1)} = 3^{2 x + 2}$ , and equal bases force equal exponents: $3 x = 2 x + 2$ , so $x = 2$ . Check: $2 7^{2} = 729$ and $9^{3} = 729$ .

Problem 8Multiple choice

A quantity grows exponentially from 300 to 1200 in 2 years. By what percentage does it grow each year?

A150%
B100%
C200%
D50%

Show the worked solution

Answer: B — 100%

Two years of growth multiply by $g^{2}$ , and $300 \cdot g^{2} = 1200$ gives $g^{2} = 4$ , so $g = 2$ . A factor of 2 means the quantity doubles each year, which is growth of $100%$ . Check: $300 \to 600 \to 1200$ . The tempting 150% comes from splitting the total rise of 300% evenly over two years, but growth compounds; it does not divide.

Problem 9Numeric answer

A machine loses $20%$ of its value every year. After how many whole years is it first worth less than half of its purchase price?

Show the worked solution

Answer: 4

The factor is $0.8$ per year, so after $t$ years the value is $0. 8^{t}$ of the original. I need the first whole $t$ with $0. 8^{t} < 0.5$ . Powers by hand: $0. 8^{2} = 0.64$ , $0. 8^{3} = 0.512$ , $0. 8^{4} = 0.4096$ . After 3 years just over half is left; after 4 years it drops below. So the answer is 4 years. Check the boundary: $0.512 > 0.5$ , so 3 is not enough.

Problem 10Multiple choice

How many real solutions does the equation $4^{x} - 2^{x} - 2 = 0$ have?

A0
B1
C2
Dinfinitely many

Show the worked solution

Answer: B — 1

Since $4^{x} = (2^{x})^{2}$ , I set $u = 2^{x}$ and get $u^{2} - u - 2 = 0$ , which factorises as $(u - 2) (u + 1) = 0$ , so $u = 2$ or $u = - 1$ . But $u = 2^{x}$ is always positive, so $u = - 1$ is impossible. Only $2^{x} = 2$ survives, giving $x = 1$ . Exactly one solution. Check: $4^{1} - 2^{1} - 2 = 0$ .

OMPT exponential functions: questions students ask

Are exponential functions on the OMPT-A?

Yes. "Exponential functions and logarithms" is one of the eight OMPT-A chapters, so roughly 4 to 6 of the 52 questions come from it, split between exponentials and logs. Expect to solve an equation like 5 times 2 to the power x equals 80 by matching bases, to turn "grows by 4% per year" into a factor of 1.04, and to evaluate or read back a model of the form b times g to the power t.

How do I solve an exponential equation without a calculator on the OMPT?

Isolate the power first, then write both sides with the same base and equate the exponents. 5 times 2 to the power x equals 80 becomes 2 to the power x equals 16, which is 2 to the power 4, so x = 4. Learn the disguises: 1/8 is 2 to the power minus 3, root 3 is 3 to the power one half, 4 to the power x is 2 to the power 2x. When the bases refuse to match, the question is a logarithm question.

What is the growth factor for a percentage decrease on the OMPT?

One minus the rate as a decimal. A decrease of 15% per year is a factor of 0.85, so after t years the quantity is the start times 0.85 to the power t. Using 0.15 is the most common wrong answer and would remove 85% each year. Two years of 15% decay lose 27.75% in total, not 30%, because the second 15% comes off a smaller amount.

Is there a difference between exponential functions on the OMPT-A and the OMPT-B?

The chapter is the same, but OMPT-B has 64 questions and adds trigonometry and integration, so exponential functions also appear inside calculus items: differentiate e to the power x or 2 to the power x, integrate one, or find a tangent to a growth curve. OMPT-B also likes equations with two different bases and the quadratic-in-disguise shape 4 to the power x minus 5 times 2 to the power x plus 4 = 0.

How do half-life questions work on the OMPT?

The quantity halves every T units of time, so the model is the start times one half to the power t over T. Count periods, not hours: 60 hours with a 20-hour half-life is 3 halvings. When you are asked for the time at which a level is reached, divide the target by the start, write that ratio as a power of one half, and read off the number of periods. If the ratio is not a power of one half, the logarithms lesson finishes the job.

Are exponential functions on the OMPT-C?

No. The OMPT-C syllabus lists Numbers, Algebra, Linear formulas and equations and Geometry. Percentage change appears as arithmetic in the Numbers chapter, so you should be able to compute a 15% decrease, but growth models with a variable in the exponent and equations like 2 to the power x equals 16 are not examined. Spend that time on exponents and roots and on linear equations.

Test hubs that cover exponential functions:OMPT-A OMPT-B OMPT-D OMPT-E OMPT-F OMPT-G All 28 topics

OMPT exponential functions practice: lesson and 10 problems

OMPT exponential functions: the lesson

What the OMPT tests in exponential functions

Notation and answer format the OMPT expects

Worked example 1: isolate, then match bases

Worked example 2: from story to number

How each OMPT variant tests exponential functions

OMPT exponential functions practice problems

OMPT exponential functions: questions students ask

Related OMPT topics and test hubs