Exam-Style Questions on Logarithms

1.	IB Analysis and Approaches

It is given that \( \log_{10} x = \frac{1}{4} \), where \( x > 0 \).

Find the value of

(a) \( \log_{10} \left(\frac{1}{x}\right) \);

(b) \( \log_{1000} x \).

Worked Solution

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2.	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:

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

Worked Solution

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3.	IB Standard

Evaluate the following, giving your answers as integers.

(a) \(\log _5 25\)

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

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

Worked Solution

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4.	IB Standard

Find the value of

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

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

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

Worked Solution

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5.	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, \(N\).

Worked Solution

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6.	IB Standard

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

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

Worked Solution

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7.	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 \(x = \frac12 \tan^{-1} 2 \).

Worked Solution

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8.	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 \( \sum_{n=1}^{16} u_n \).

Worked Solution

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9.	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 \(p\).

Worked Solution

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