Exponential and logarithmic functions  Examples

ExamplesThe 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 \(M = \frac{2}{3}\log E  2.9\). ] 
SolutionsLet \({E_0}\) 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.

ExamplesSolve the equation \({\log _4}({x^2} + 3) = {\log _4}x + 1\). (Give your answer correct to 3 significant figures if necessary.)

SolutionsAssume 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 \(x = \underline{\underline {{\rm{ }}1{\rm{ }}}} \)

ExamplesThe 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) \({\log _{\frac{1}{7}}}x \ge  0.3\) 
Solutions
(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.).