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Exam-Style Questions on Logarithms

Problems on Logarithms adapted from questions set in previous Mathematics exams.

1.

IB Standard

Find the value of the following:

(a) \(log_464\);

(b) \(log_7\frac17\);

(c) \(log_{25}5\);

(d) Use the solutions to the previous parts of this question to help solve:

$$log_464+log_7\frac17-log_{25}5=log_4x$$

2.

IB Standard

Evaluate the following, giving your answers as integers.

(a) \(\log _5 25\)

(b) \(\log _6 3 + \log _6 12\)

(c) \(\log _2 12 - \log _2 6\)


3.

IB Standard

Find the value of

(a) \(\log _4 2 + \log _4 8\)

(b) \(\log_2 60-\log_2 15\)

(c) \(27^{\log_3 4}\)


4.

IB Applications and Interpretation

In an old science fiction book the author described the intensity of reverse polarity, \(P\) measured in treckons, is a function of the nebula thrust, \(N\) measures in whovians. The intensity level is given by the following formula.

$$P = 7 \log_{10}(N \times 10^{8}), N \ge 0$$

(a) An space shuttle has a nebula thrust of \(9.1 × 10^{-3}\) whovians. Calculate the intensity level, \(P\) of the shuttle.

(b) A different space shuttle has an intensity level of 112 trekons. Find its nebula thrust, \(N\).

Flying Saucer

5.

IB Standard

(a) Solve \(4x^2 - 8x - 5 = 0\)

(b) Hence solve \(4 \times 25^x - 8 \times 5^x = 5\)


6.

IB Analysis and Approaches

(a) Show that \( \log_4 (\sin 2x +2) = \log_2 \sqrt{\sin 2x + 2 }\)

(b) Hence or otherwise solve \( \log_2 (2 \cos x) = \log_4 (\sin 2x + 2) \) to show that \(x = \frac12 \tan^{-1} 2 \).


7.

IB Standard

An arithmetic sequence has \(u_1 = \log_h(j)\) and \(u_2 = \log_h(jk)\), where \(h > 1\) and \(j, k \gt 0\).

(a) Show that \(d = \log_h(k)\).

(b) Let \(j = h^5\) and \(k = h^7\). Find the value of \( \sum_{n=1}^{16} u_n \).


8.

IB Standard

Consider the function \(f (x) = \log_p(24x - 18x^2)\) , for \(0 \lt x \lt 1\), where \(p \gt 0\).

The equation \(f (x) = 3\) has exactly one solution. Find the value of \(p\).


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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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