### Exponential and logarithmic functions - Examples

• The magnitude of an earthquake is 6.8 in the Richter scale, and the energy released from an subsequent earthquake is 1% of that released from the first earthquake. Find the magnitude of the subsequent earthquake. (Give your answer correct to 1 decimal place.)
[ The relation between the magnitude (M) of an earthquake and the energy released (E units) from the earthquake can be expressed as $$M = \frac{2}{3}\log E - 2.9$$. ]
• Let $${E_0}$$ units and E units be the energy released from the first earthquake and that released from the subsequent earthquake respectively.
$$\begin{array}{1}6.8 = \frac{2}{3}\log {E_0} - 2.9\\\frac{2}{3}\log {E_0} = 9.7\\\log {E_0} = 14.55\\{E_0} = {10^{14.55}}\end{array}$$
∵ $$E = {E_0} \times \frac{1}{{100}}$$
$$\begin{array}{1}∴E = {10^{14.55}} \times \frac{1}{{100}}\\ = {10^{14.55}} \times {10^{ - 2}}\\ = {10^{12.55}}\end{array}$$
Magnitude of the subsequent earthquake
$$\begin{array}{l} = \frac{2}{3}\log {10^{12.55}} - 2.9\\ = \frac{2}{3} \times 12.55 - 2.9\end{array}$$
$$= 5.5$$ (corr. to 1 d.p.)
∴ The magnitude of the subsequent earthquake is 5.5 in the Richter scale.
• Solve the equation $${\log _4}({x^2} + 3) = {\log _4}x + 1$$. (Give your answer correct to 3 significant figures if necessary.)
• Assume that $${x^2} + 3 > 0$$ and $$x > 0$$.
$$\begin{array}{1}{\log _4}({x^2} + 3) = {\log _4}x + 1\\{\log _4}({x^2} + 3) = {\log _4}x + {\log _4}4\\{\log _4}({x^2} + 3) = {\log _4}4x\end{array}$$
$$\begin{array}{1}{x^2} + 3 = 4x\\{x^2} - 4x + 3 = 0\\(x - 3)(x - 1) = 0\end{array}$$
$$x = 3$$ or $$x = 1$$
When $$x = 3$$,
$$\begin{array}{1}{x^2} + 3 = {3^2} + 3\\ = 9 + 3\\ = 12\\ > 0\end{array}$$ and $$\begin{array}{c}4x = 4{(3)^{}}\\ = 12\\ > 0\\\end{array}$$
When $$x = 1$$,
$$\begin{array}{1}{x^2} + 3 = {1^2} + 3\\ = 1 + 3\\ = 4\\ > 0\end{array}$$ and $$\begin{array}{c}4x = 4{(1)^{}}\\ = 4\\ > 0\\\end{array}$$
∵ $$x = \underline{\underline 3}$$ or $$x = \underline{\underline {{\rm{ }}1{\rm{ }}}}$$
• The figure below shows the graph of $$y = {\log _{\frac{1}{7}}}x$$.

(a) Using the given graph, find the values of the following. (Give your answers correct to 1 decimal place if necessary.)
(i) $${\log _{\frac{1}{7}}}0.8$$
(ii) $${\log _{\frac{1}{7}}}1.5$$
(iii) $${\log _{\frac{1}{7}}}3.2$$
(b) Solve the following inequalities graphically. (Give your answers correct to 1 decimal place if necessary.)
(i) $${\log _{\frac{1}{7}}}x < 0.6$$
(ii) $${\log _{\frac{1}{7}}}x \ge - 0.3$$

• (a) From the graph,
(i) $${\log _{\frac{1}{7}}}0.8 = \underline{\underline {0.1}}$$ (corr. to 1 d.p.)
(ii) $${\log _{\frac{1}{7}}}1.5 = \underline{\underline { - {\rm{ }}0.2}}$$ (corr. to 1 d.p.)
(iii) $${\log _{\frac{1}{7}}}3.2 = \underline{\underline { - {\rm{ }}0.6}}$$ (corr. to 1 d.p.)

(b) From the graph,(with reference to the lines each added above
(i) the solutions of the inequality $${\log _{\frac{1}{7}}}x < 0.6$$ are $$x > 0.3$$ (corr. to 1 d.p.).
(ii)the solutions of the inequality $${\log _{\frac{1}{7}}}x \ge - 0.3$$ are $$0 < x \le 1.8$$ (corr. to 1 d.p.).