OMPT drill — Functions

OMPT logarithms practice: lesson and 10 problems

Log rules, change of base, and solving logarithmic equations. In my sessions this is the topic with the widest gap between "seen it" and "can do it under time". 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 logarithms: the lesson

What the OMPT tests in logarithms

Logarithms sit in the chapter "Exponential functions and logarithms" on OMPT-A, B, D, E, F and G. On OMPT-A that chapter is one of eight, so roughly five to seven of the 52 questions come from it, about half of them on logs. The sub-skills come in four families. Evaluating a log from the definition ( $lo g_{2} 32$ , $lo g_{5} \frac{1}{25}$ ). Rewriting with the three rules, in both directions: expanding one log into several, or condensing several into one. Solving an equation with the unknown inside a log, which ends with a domain check. And solving an equation with the unknown in the exponent, $2^{x} = 10$ or $1.0 5^{t} = 2$ , where the log is the tool that brings the exponent down.

The heavy families are the last two. A typical question reads "Solve $lo g_{3} x + lo g_{3} (x - 2) = 1$ " or "After how many years has the investment doubled?", and both hide the same trap: a solution the algebra produces and the logarithm refuses. On OMPT-D and OMPT-F the log step usually sits inside a longer question about a growth model or a derivative.

Notation and answer format the OMPT expects

$lo g_{b} a$ means "the power of $b$ that gives $a$ ". $ln x$ is $lo g_{e} x$ and $lo g x$ with no base is base 10. The argument must be positive and the base must be positive and not equal to 1.
Exact answers are expected. Type $\frac{3}{2}$ as 3/2, and an answer such as $\frac{l n 10}{l n 2}$ as ln(10)/ln(2) or log_2(10); the platform accepts equivalent exact forms. Give a decimal only when the question says "round to".
No calculator unless the question supplies one. If it quotes $lo g 2 \approx 0.301$ , use that value.
When an equation has two solutions, give both; when the domain kills every candidate, the answer is "no solution", and the box accepts that.

Worked example 1: evaluating from the definition

Evaluate $lo g_{2} 32$ , $lo g_{5} \frac{1}{25}$ and $lo g_{4} 8$ .
$lo g_{2} 32$ : 2 to what power is 32? Count the doublings, 2, 4, 8, 16, 32, five of them, so the answer is 5.
$lo g_{5} \frac{1}{25}$ : $\frac{1}{25} = 5^{- 2}$ , so the answer is $- 2$ . A log can be negative; its argument cannot.
$lo g_{4} 8$ resists guessing, so change to base 2: $\frac{l o g _{2} 8}{l o g _{2} 4} = \frac{3}{2}$ .
Check the last one: $4^{3/2} = (4)^{3} = 2^{3} = 8$ . A fractional log is legal and the OMPT uses this exact shape to see whether you reach for the definition or panic.

Worked example 2: combine, convert, check

Solve $lo g_{3} x + lo g_{3} (x - 2) = 1$ .
Domain first: both arguments must be positive, so $x > 2$ . Write that down before any algebra.
Product rule: $lo g_{3} (x (x - 2)) = 1$ . Definition: $x (x - 2) = 3^{1} = 3$ .
Solve the quadratic: $x^{2} - 2 x - 3 = 0$ , so $(x - 3) (x + 1) = 0$ , giving $x = 3$ or $x = - 1$ .
Compare with the domain: $x = - 1$ would need $lo g_{3} (- 1)$ , which does not exist. Reject it. The only solution is $x = 3$ , and a quick check gives $lo g_{3} 3 + lo g_{3} 1 = 1 + 0 = 1$ .

The full lesson states the three rules and shows where they come from, then works through exponent equations without a matching base, expanding and condensing, a quotient-rule equation, a quadratic in $lo g_{2} x$ , a doubling-time word problem and a "how many solutions" question. It closes with the edge cases the OMPT likes and the slips I mark most often.

How each OMPT variant tests logarithms

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 3 to 4 of 52 questions, out of the 5 to 7 for the whole exponentials and logarithms chapter
Official chapter: Exponential functions and logarithms, 10% of the paper.
Evaluate a log from the definition, rewrite with the rules, solve a log equation with one rejected candidate, and solve an exponential equation by taking a log. Exact answers such as log_2(10) or 3/2.
OMPT-B
64 questions
Testedroughly 3 to 5 of 64 questions
Official chapter: Exponential functions and logarithms, 10% of the paper.
Same shapes as OMPT-A plus natural logs feeding the differentiation and integration chapters, so ln appears as a step inside a derivative or an integral.
OMPT-C
30 questions
Not tested
OMPT-C covers numbers, algebra, linear equations and geometry, with no exponential or logarithm chapter. Spend the time on linear equations and geometry 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.
Long multi-part questions where a log solves for time in a growth model or unlocks an equation before a derivative or a limit is taken. Exact forms expected.
OMPT-E
40 questions
Testedroughly 2 to 4 of 40 questions
Official chapter: Exponential functions and logarithms, 5% of the paper.
Short numeric answers: evaluate, apply the rules, solve a log equation, and solve an exponential equation for time. No calculus follows, so the questions stay self-contained.
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: doubling times, compound growth solved for the number of periods, and a log step inside a derivative question. Answers are exact or rounded as the question specifies.
OMPT-G
26 questions
Testedroughly 2 to 3 of 26 questions
Official chapter: Functions, graphs and equations, 45% of the paper.
Evaluation, rules and exponential equations solved with a log, sometimes inside a sequences question about geometric growth or a chemistry item about pH-style scales.

OMPT logarithms 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

Evaluate $lo g_{3} 81$ .

Show the worked solution

Answer: 4

The logarithm asks: 3 to which power gives 81? Since $3^{4} = 81$ , the answer is 4. When I blank on a log, I rewrite it as its exponential twin: $lo g_{3} 81 = x$ means $3^{x} = 81$ .

Problem 2Multiple choice

For positive numbers $a$ and $b$ , which expression equals $lo g a + lo g b$ ?

A $lo g (ab)$
B $lo g (a + b)$
C $lo g a \cdot lo g b$
D $lo g (a^{b})$

Show the worked solution

Answer: A — $lo g (ab)$

Adding logs multiplies their arguments: $lo g a + lo g b = lo g (ab)$ . This mirrors the exponent rule $1 0^{x} \cdot 1 0^{y} = 1 0^{x + y}$ — logs turn multiplication into addition, which is the whole reason they were invented.

Problem 3Numeric answer

Solve for $x$ : $lo g_{2} (x - 1) = 5$ .

Show the worked solution

Answer: 33

I convert to exponential form: $x - 1 = 2^{5} = 32$ , so $x = 33$ . The argument is positive ( $33 - 1 = 32 > 0$ ), so the solution is valid. That domain check takes two seconds and catches phantom solutions in harder problems.

Problem 4Multiple choice

Given $lo g 2 \approx 0.301$ , what is $lo g 8$ approximately?

A $0.903$
B $0.602$
C $1.204$
D $0.027$

Show the worked solution

Answer: A — $0.903$

Since $8 = 2^{3}$ , the power rule gives $lo g 8 = 3 lo g 2 \approx 3 \cdot 0.301 = 0.903$ . No calculator needed once you spot that 8 is a power of 2.

Problem 5Spot the error

A student simplifies. Step 1: $lo g (x + y) = lo g x + lo g y$ . Step 2: with $x = 10$ , $y = 90$ , the left side becomes $lo g 100 = 2$ . Step 3: the right side becomes $1 + lo g 90 \approx 2.95$ . Which step contains the error?

Show the worked solution

Answer: Step 1

Step 1 states a rule that does not exist: the log of a sum does not split. Steps 2 and 3 actually expose the problem, since the two sides disagree ( $2 \neq = 2.95$ ). The genuine rule concerns products: $lo g (x y) = lo g x + lo g y$ , and indeed $lo g (10 \cdot 90) = lo g 900 \approx 2.95$ matches the right side.

Problem 6Numeric answer

Solve for $k$ : $1 0^{2 k} = 1000$ .

Show the worked solution

Answer: 1.5

Taking $lo g_{10}$ of both sides — or simply writing $1000 = 1 0^{3}$ — gives $2 k = 3$ , so $k = 1.5$ . Check: $1 0^{3} = 1000$ .

Problem 7Multiple choice

Let $lo g_{b} 2 = p$ and $lo g_{b} 3 = q$ for some base $b > 1$ . Which expression equals $lo g_{b} 12$ ?

A $2 p + q$
B $p + 2 q$
C $2 pq$
D $p^{2} q$

Show the worked solution

Answer: A — $2 p + q$

I factorise the argument into the primes I have logs for: $12 = 2^{2} \cdot 3$ . The product rule splits it, $lo g_{b} 12 = lo g_{b} (2^{2}) + lo g_{b} 3$ , and the power rule brings the 2 down: $2 lo g_{b} 2 + lo g_{b} 3 = 2 p + q$ . Check with $b = 10$ : $2 (0.301) + 0.477 = 1.079$ , and $lo g 12 \approx 1.079$ .

Problem 8Numeric answer

Solve for the base $x$ : $lo g_{x} 125 = 3$ .

Show the worked solution

Answer: 5

The definition says: $x$ to the power 3 gives 125, so $x^{3} = 125$ and $x = 5$ . A base must be positive and not 1, and 5 qualifies. Check: $5^{3} = 125$ . The unknown can sit in any of the three positions of $lo g_{b} a = c$ , and the exponential twin $b^{c} = a$ handles all three.

Problem 9Numeric answer

The equation $(lo g_{3} x)^{2} = lo g_{3} (x^{2}) + 3$ has two solutions. Give their product.

Show the worked solution

Answer: 9

I rewrite $lo g_{3} (x^{2})$ as $2 lo g_{3} x$ with the power rule, then substitute $t = lo g_{3} x$ : $t^{2} = 2 t + 3$ , so $t^{2} - 2 t - 3 = 0$ and $(t - 3) (t + 1) = 0$ . Back-substituting, $lo g_{3} x = 3$ gives $x = 27$ , and $lo g_{3} x = - 1$ gives $x = \frac{1}{3}$ . Both are positive, so both survive, and their product is $27 \cdot \frac{1}{3} = 9$ . Check $x = \frac{1}{3}$ : left side $(- 1)^{2} = 1$ , right side $- 2 + 3 = 1$ .

Problem 10Numeric answer

A savings account grows by 6% per year. After how many whole years does the balance first exceed twice the deposit? Use $ln 2 \approx 0.693$ and $ln 1.06 \approx 0.0583$ .

Show the worked solution

Answer: 12

The balance after $t$ years is the deposit times $1.0 6^{t}$ , so doubling means $1.0 6^{t} = 2$ ; the deposit cancels. Taking $ln$ of both sides and bringing the exponent down: $t = \frac{l n 2}{l n 1.06} \approx \frac{0.693}{0.0583} \approx 11.9$ . The balance passes double between year 11 and year 12, so the first whole year is 12. Check: $1.0 6^{11} \approx 1.90$ and $1.0 6^{12} \approx 2.01$ .

OMPT logarithms: questions students ask

Are logarithms on the OMPT-A?

Yes. OMPT-A has a chapter called "Exponential functions and logarithms", one of its eight chapters. Expect roughly three or four log questions out of 52: evaluating a log, rewriting with the product, quotient and power rules, solving an equation with the unknown inside a log, and solving an exponential equation by taking a log of both sides. Only OMPT-C skips the chapter.

How do I solve logarithmic equations on the OMPT?

State the domain first: every argument must be positive. Condense each side to a single log with the rules, then convert with the definition, log base b of a equals c means b to the power c equals a. Solve the resulting equation, usually a linear or a quadratic, and reject any candidate outside the domain. The OMPT plants a rejected candidate in most log equations, so the last step is where the mark is.

Do I need a calculator for logarithms on the OMPT?

No. The maths variants have no calculator, and log questions are built so that exact answers are typed: log_2(10), ln(10)/ln(2), 3/2. When a decimal is needed the question quotes the values you need, such as log 2 is about 0.301, and expects you to combine them with the rules. Do not try to estimate logs from memory beyond the anchor values log_b 1 = 0 and log_b b = 1.

What is the difference between ln and log on the OMPT?

ln x is the natural logarithm, base e, and log x with no base written is base 10. They obey the same three rules and differ only by a constant factor through change of base. Calculus questions on OMPT-B, D and F prefer ln because the derivative of ln x is 1/x. The base-10 log appears when a question hands you an approximate value to work with.

How many solutions does a log equation have on the OMPT?

Often fewer than the algebra suggests. Combining logs and converting usually produces a quadratic with two candidates, and the domain then rejects one of them, or both. The OMPT asks "how many solutions" precisely for this reason. Count only the candidates that keep every argument positive, and be ready to answer one, or none.

How do I solve 2 to the power x equals 10 without a calculator?

Take a logarithm of both sides and bring the exponent down with the power rule: x times ln 2 equals ln 10, so x equals ln 10 divided by ln 2, which is also log base 2 of 10. That exact form is the answer the OMPT wants. A sanity check is free: 2 cubed is 8 and 2 to the fourth is 16, so x is a little above 3.

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

OMPT logarithms practice: lesson and 10 problems

OMPT logarithms: the lesson

What the OMPT tests in logarithms

Notation and answer format the OMPT expects

Worked example 1: evaluating from the definition

Worked example 2: combine, convert, check

How each OMPT variant tests logarithms

OMPT logarithms practice problems

OMPT logarithms: questions students ask

Related OMPT topics and test hubs