# 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

(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, $$N$$.

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

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

### 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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