Tag: logarithm

Hardest Questions in Additional Mathematics (A Math)

Additional Mathematics questions can range from standard all the way to super challenging among the secondary schools in Singapore.

Certain schools (such as IP schools), and also some schools such as Anderson, Chung Cheng High School, are well known for setting hard A Math papers.

Note that even though top schools set hard A Math papers, it is not often the case that top schools teach or prepare well their students for the tests! Often, the teachers in school teach at a basic level (due to time constraints or other factors), but still test at an advanced level. Hence, many students in top IP schools are not well prepared for their school’s tests (unless they have excellent self study skills or have a parent or tutor to guide them). It is not uncommon for a student in a top IP school to be failing his/her math tests due to the above phenomena (difficult tests which do not match what is taught in school).

Some of the more difficult types of questions in the A Math syllabus are listed below.

Algebra

  1. Conditions for ax^2 + bx + c to be always positive (or always negative).
    This type of question has potential to be very tricky. Somehow, many students will assume wrongly that b^2-4ac is always positive as well (where it should be the opposite).
  2. Partial Fractions with Improper Fractions.
    Only top schools tend to test improper partial fractions. Many students will miss out long division or make mistakes along the way.
  3. Binomial Theorem.
    Many students have serious problems with this topic. Also, not many seem to know that {n\choose 2}=\frac{n(n-1)}{2}.

Logarithm

Trigonometry

  • Sketching of Tangent graphs.
    90% of all sketching questions are on Sine or Cosine. Only top schools will set tangent sketching questions, and many students will be caught unaware.
  • Half-angle formula sin(x/2) or Quadruple angle formula sin(4x)
    Top schools like to test half-angle formula, many students who have not seen such questions will be stuck.

Integration

  • Finding area to the left of the curve, i.e. \int x\,dy.
    Most schools kind of brush off this type of questions during teaching. But it is a hot topic for testing among top schools. Hence, students will have a hard time solving it if they lack practice for this type of questions.

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Free Logarithm Worksheet for Secondary School

Logarithm Worksheet (PDF): Logarithm Worksheet

Comprises of 9 typical Logarithm questions of varying difficulty, with answer provided.

Logarithm is a popular question topic for Additional Mathematics that appear in Secondary 3 or Secondary 4 A Maths. Most students have some problems with this topic initially due to its newness and also the types of challenging questions that can appear in this topic.

Logarithm can also be easily connected and interrelated to topics such as:

  • Indices
  • Simultaneous Equations
  • Linear Law

Do also check out this highly challenging Logarithm question:

O Level Logarithm Question (Challenging)

Singapore Math (High School): Logarithm Question

The following is a follow up video on my earlier post on Logarithm Question (Challenging).

singapore-math-logarithmThe video is posted on: http://mathtuition88.blogspot.sg/2014/12/singapore-math-high-school-logarithm.html

Thanks for watching!

Featured Book:

John Napier: Life, Logarithms, and Legacy

Featured Posts:

Recommended Singapore Math Books

Logarithm rules and Indices rules

Two page helpsheet and formula list regarding Logarithm rules and Indices rules, with common mistakes to avoid:

Logarithm and Indices Rules

Notes on Logarithm by NUS

Notes on Logarithm by NUS:

Source: http://www.math.nus.edu.sg/~matngtb/Calculus/mathcentre/mathcentre_workbooks/web-logarithms-new-july03.pdf

Quote:

Logarithms appear in all sorts of calculations in engineering and science, business and economics.

Before the days of calculators they were used to assist in the process of multiplication by replacing the operation of multiplication by addition. Similarly, they enabled the operation of division to be replaced by subtraction. They remain important in other ways, one of which is that they provide the underlying theory of the logarithm function. This has applications in many fields, for example, the decibel scale in acoustics.

In order to master the techniques explained here it is vital that you do plenty of practice exercises so that they become second nature.

Read more at: http://www.math.nus.edu.sg/~matngtb/Calculus/mathcentre/mathcentre_workbooks/web-logarithms-new-july03.pdf

Logarithm and Exponential Question: A Maths Question

Question:

Solve (4x)^{\lg 5} = (5x)^{\lg 7}

Solution:

4^{\lg 5}\cdot x^{\lg 5}=5^{\lg 7}\cdot x^{\lg 7}

\displaystyle\frac{4^{\lg 5}}{5^{\lg 7}}=\frac{x^{\lg 7}}{x^{\lg 5}}=x^{\lg 7-\lg 5}

Using calculator, and leaving answers to at least 4 s.f.,

0.6763=x^{0.1461}

Lg both sides,

\lg 0.6763=0.1461\lg x

\lg x=\frac{\lg 0.6763}{0.1461}=-1.1626

x=10^{-1.1626}=0.0688 (3 s.f.)

Check answer (to prevent careless mistakes):

LHS=(4\times 0.0688)^{\lg 5}=0.406

RHS=(5\times 0.0688)^{\lg 7}=0.406

Since LHS=RHS, we have checked that our answer is valid.

O Level Logarithm Question (Challenging)

Question:

Given \displaystyle\log_9{a} = \log_{12}{b} =\log_{16}{(a+b)}, find the value of  \displaystyle\frac{a}{b}.

Solution:

Working with logarithm is tricky, we try to transform the question to an exponential question.

Let \displaystyle y=\log_9{a} = \log_{12}{b} =\log_{16}{(a+b)}

Then, we have a=9^y=3^{2y}

b=12^y=3^y\cdot 2^{2y}

a+b=16^y=2^{4y}.

Here comes the critical observation:

Observe that \boxed{a(a+b)=b^2}.

Divide throughout by b^2, we get \displaystyle (\frac{a}{b})^2+\frac{a}{b}=1.

Hence, \displaystyle (\frac{a}{b})^2+\frac{a}{b}-1=0.

Solving using quadratic formula (and reject the negative value since a and b has to be positive for their logarithm to exist),

We get \displaystyle\frac{a}{b}=\frac{-1+\sqrt{5}}{2}.

If you have any questions, please feel free to ask me by posting a comment, or emailing me.

(I will usually explain in much more detail if I teach in person, than when I type the solution)