Mathematics Admissions Test (MAT) Exponentials, Logarithms and Sequences Syllabus

Exponentials, Logarithms and Sequences syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exponentials, Logarithms and Sequences in Mathematics Admissions Test (MAT), not a summary of it.

1. 1

   Exponentials and Logarithms

   3 topics

   • Exponential Functions
     • Graphs and properties of a to the x for a greater than 1 and between 0 and 1
     • Growth and decay behaviour
   • Laws of Logarithms
     • Product, quotient and power rules
     • Change of base and relationship to indices
   • Solving Exponential and Log Equations
     • Taking logs to solve a to the x equals b
     • Equations that reduce to quadratics in an exponential term
     • Domain restrictions from the logarithm
2. 2

   Sequences

   3 topics

   • Arithmetic Sequences and Series
     • nth term and sum to n terms
     • Problems with unknown first term and common difference
   • Geometric Sequences and Series
     • nth term, finite sum and the sum to infinity
     • Convergence condition on the common ratio
   • Recurrence and Iteration
     • Sequences defined recursively
     • Detecting periodic or fixed-point behaviour
     • Conjecturing a closed form from terms
3. 3

   Summation and Series Reasoning

   3 topics

   • Sigma Notation
     • Reading, writing and manipulating sums in sigma notation
     • Splitting and shifting summation indices
   • Standard Summation Results
     • Sums of the first n integers and squares
     • Telescoping sums
   • Series in Problem Solving
     • Combining arithmetic and geometric reasoning in unfamiliar contexts

Exponentials, Logarithms and Sequences flashcards for Mathematics Admissions Test (MAT)

23 of 49 cards from the Exponentials, Logarithms and Sequences deck — real questions with worked answers.

1. What is the general form of an exponential function, and what condition must the base satisfy?

   $f(x) = a^{x}$, where $a > 0$ and $a \neq 1$. The constant $a$ is the base and $x$ is the exponent.

2. What is the natural exponential function, and approximately what value is its base?

   $f(x) = e^{x}$, where $e \approx 2.71828$ (Euler's number).

3. What is the key calculus property that makes $e^{x}$ special?

   It is its own derivative: $\frac{d}{dx}e^{x} = e^{x}$. More generally $\frac{d}{dx}e^{kx} = ke^{kx}$.

4. For the exponential function $a^{x}$ with $a>1$, describe its behaviour as $x \to \infty$ and $x \to -\infty$.

   As $x \to \infty$, $a^{x} \to \infty$ (growth). As $x \to -\infty$, $a^{x} \to 0^{+}$, so the $x$-axis is a horizontal asymptote ($y=0$).

5. What point does every graph $y = a^{x}$ pass through, and why?

   It passes through $(0,1)$, because $a^{0} = 1$ for any valid base $a$.

6. What distinguishes exponential growth from exponential decay in $y = a^{x}$?

   Growth occurs when $a > 1$ (function increases); decay occurs when $0 < a < 1$ (function decreases).

7. How is the logarithm $\log_{a} x$ defined in terms of exponentials?

   $\log_{a} x = y$ means exactly $a^{y} = x$. The logarithm is the inverse of the exponential with base $a$ (for $x>0$, $a>0$, $a\neq 1$).

8. What is the natural logarithm and what is its base?

   $\ln x = \log_{e} x$, the logarithm with base $e$. It is the inverse of $e^{x}$, so $\ln(e^{x}) = x$ and $e^{\ln x} = x$.

9. State the three core laws of logarithms (product, quotient, power).

   $\log_{a}(xy) = \log_{a} x + \log_{a} y$; $\quad \log_{a}\!\left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$; $\quad \log_{a}(x^{k}) = k\log_{a} x$.

10. What are the values of $\log_{a} 1$ and $\log_{a} a$?

    $\log_{a} 1 = 0$ (since $a^{0}=1$) and $\log_{a} a = 1$ (since $a^{1}=a$).

11. State the change of base formula for logarithms.

    $\log_{a} x = \dfrac{\log_{b} x}{\log_{b} a}$ for any valid base $b$. Commonly $\log_{a} x = \dfrac{\ln x}{\ln a}$.

12. How can $\log_{a}\!\left(\frac{1}{x}\right)$ be rewritten using log laws?

    $\log_{a}\!\left(\frac{1}{x}\right) = \log_{a}(x^{-1}) = -\log_{a} x$.

13. What is the reciprocal (base-switch) relationship $\log_{a} b$ in terms of $\log_{b} a$?

    $\log_{a} b = \dfrac{1}{\log_{b} a}$.

14. What is the general method for solving an exponential equation such as $a^{x} = b$?

    Take logarithms of both sides: $x \ln a = \ln b$, so $x = \dfrac{\ln b}{\ln a} = \log_{a} b$.

15. Solve $2^{x} = 10$, giving the exact answer.

    $x = \log_{2} 10 = \dfrac{\ln 10}{\ln 2} \approx 3.322$.

16. What substitution turns the equation $e^{2x} - 5e^{x} + 6 = 0$ into a quadratic, and what is it?

    Let $u = e^{x}$ (so $e^{2x}=u^{2}$), giving $u^{2} - 5u + 6 = 0$, i.e. $(u-2)(u-3)=0$, so $e^{x}=2$ or $e^{x}=3$.

17. When solving $\log$ equations, why must solutions be checked at the end?

    Because logarithms require positive arguments; any candidate solution that makes the argument of a log zero or negative must be rejected as extraneous.

18. Solve $\log_{2}(x) + \log_{2}(x-2) = 3$.

    Combine: $\log_{2}[x(x-2)] = 3 \Rightarrow x(x-2)=8 \Rightarrow x^{2}-2x-8=0 \Rightarrow (x-4)(x+2)=0$. Reject $x=-2$ (negative argument), so $x=4$.

19. Define an arithmetic sequence and its common difference.

    A sequence in which each term differs from the previous by a constant. The common difference is $d = a_{n+1} - a_{n}$.

20. State the formula for the $n$-th term of an arithmetic sequence with first term $a$ and common difference $d$.

    $a_{n} = a + (n-1)d$.

21. State the formula for the sum of the first $n$ terms of an arithmetic series.

    $S_{n} = \dfrac{n}{2}\big(2a + (n-1)d\big) = \dfrac{n}{2}(a + l)$, where $l$ is the last term.

22. Find the sum of the first 100 positive integers using the arithmetic series formula.

    $S_{100} = \dfrac{100}{2}(1 + 100) = 50 \times 101 = 5050$.

23. Define a geometric sequence and its common ratio.

    A sequence in which each term is a constant multiple of the previous one. The common ratio is $r = \dfrac{a_{n+1}}{a_{n}}$.

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