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Exam-Style Questions.

Problems adapted from questions set for 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 Standard

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

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


5.

IB Standard

Part of the graph of \(f(x) = {\log _b}(x + 4)\) for \(x > - 4\) is shown below.

Graph

The graph passes through A(4, 3) , has an x-intercept at (-3, 0) and has an asymptote at \(x = - 4\).

(a) Find the value of \(b\).

The graph of \(f(x)\) is reflected in the line \(y = x\) to give the graph of \(g(x)\).

(b) Write down the y-intercept of the graph of \(g(x)\).

(c) Sketch the graph of \(g(x)\), noting clearly any asymptotes and the image of A.

(d) Find \(g(x)\) in terms of \(x\).


6.

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 the common difference, \(d = \log_h(k)\).

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


7.

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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