Exponential and logarithmic functions  (3.43.7)
Understand the properties of exponential functions and logarithmic functions and recognise the features of their graphs

Graph
\(f(x)=log_{a}x\)

Logarithmic Graph\({\rm{y}} = {\log _{\rm{a}}}x\)
 the graph of the logarithmic function y and the graph of the corresponding exponential function
\({\rm{y}} = {{\rm{a}}^{\rm{x}}}\) are mirror images of each other, with the line \({\rm{y}} = {\rm{x}}\) as the reflectional axis.
 the graph always intersects at ( 1 , 0 )
 Case i) \({\rm{a}} > 1\)
o When \(0 < {\rm{x}} < 1{\rm{\;}} \to {\rm{\;y}} < 0\)
o When \({\rm{x}} > 1{\rm{\;\;}} \to {\rm{y}} > 0\)
o For any value of x, value of y increases as x increases
 Case ii) \({\rm{a}} < 1\)
o When \(0 < {\rm{x}}\left\langle {1{\rm{\;}} \to {\rm{\;y}}} \right\rangle 0\)
o When \({\rm{x}} > 1{\rm{\;\;}} \to {\rm{y}} < 0\)
o For any value of x, value of y decreases as x increases 
Exponential Graph\({\rm{y}} = {{\rm{a}}^{\rm{x}}}\)
 the graph of the logarithmic function y and the graph of the corresponding exponential function
\({\rm{y}} = {{\rm{a}}^{\rm{x}}}\) are mirror images of each other, with the line \({\rm{y}} = {\rm{x}}\) as the reflectional axis.
 the graph always intersects at ( 0 , 1 )
 Case i) \({\rm{a}} > 1\)
o When \({\rm{x}} < 0 \to {\rm{\;y}} < 1\)
o When \({\rm{x}} > 0 \to {\rm{\;y}} > 1\)
o For any value of x, value of y increases as x increases
 Case ii) \(a < 1\)
o When \({\rm{x}}\left\langle {0 \to {\rm{\;y}}} \right\rangle 1\)
o When \({\rm{x}} > 0 \to {\rm{\;y}} < 1\)
o For any value of x, value of y decreases as x increases
Solve exponential equations and logarithmic equations

ExamplesSolve the following equations. (Give your answers correct to 2 decimal places if necessary.)
(a) \({6^{x{\rm{ }} + {\rm{ }}2}} = {3^{3x{\rm{ }}  {\rm{ }}4}}\)
(b) \(\log (x + 5)  \log (2x  1) = \log 6\)

Solutions\(\begin{array}{1}(a)\qquad{6^{x{\rm{ }} + {\rm{ }}2}} = {3^{3x{\rm{ }}  {\rm{ }}4}}\\\log {6^{x{\rm{ }} + {\rm{ }}2}} = \log {3^{3x{\rm{ }}  {\rm{ }}4}}\\(x + 2)\log 6 = (3x  4)\log 3\\x\log 6 + 2\log 6 = 3x\log 3  4\log 3\\3x\log 3  x\log 6 = 2\log 6 + 4\log 3\\x = \frac{{2\log 6 + 4\log 3}}{{3\log 3  \log 6}}\end{array}\)
\( = \underline{\underline {5.30}} \) (corr. to 2 d.p.)
\(\begin{array}{1}(b)\qquad\log (x + 5)  \log (2x  1) = \log 6\\\log \frac{{x + 5}}{{2x  1}} = \log 6\\\frac{{x + 5}}{{2x  1}} = 6\\x + 5 = 6(2x  1)\\x + 5 = 12x  6\\11x = 11\\x = \underline{\underline {{\rm{ }}1{\rm{ }}}} \end{array}\)
Appreciate the applications of logarithms in reallife situations

Theory Logarithms are widely applicable in different physics, mathematics and engineering areas.\({10^{  12}}\;{\bf{W}}{{\bf{m}}^{  2}}\)
For example, to interpret the loudness of sounds from intensity of the sound, the following formula is applicable
\({\bf{D}} = 10{\bf{log}}\left( {\frac{{\bf{I}}}{{{{\bf{I}}_{\bf{o}}}}}} \right)\)
 where I is the intensity of the sound, measured in\({\bf{W}}{{\bf{m}}^{  2}}\) (units are not required, will be given in question if necessary)
 \({{\bf{I}}_{\bf{o}}}\) is the threshold of hearing, \({10^{  12}}\;{\bf{W}}{{\bf{m}}^{  2}}\)
 \({\bf{D}}\;\)is the loudness of sound, measured in dB.

Examples(a) Given that the intensity of the noise generated by a small explosion is \(7 \times {10^{  2}}\) unit, find the loudness of the noise
in dB. (Give your answer correct to the nearest integer.)
(b) The energy released from an earthquake was \(6 \times {10^{15}}\) units and the energy released from a subsequent earthquake
was \(9 \times {10^7}\) units. What is the difference in magnitude between these two earthquakes in the Richter scale?(Give your
answer correct to 1 decimal place.) 
Solutions(a) Loudness of the noise
\(\begin{array}{l} = 10\,\log [{10^{12}} \times (7 \times {10^{  2}})]{\rm{ dB}}\\ = {\rm{10}}\,{\rm{log(}}7 \times {10^{10}}{\rm{) dB}}\end{array}\)
\( = \underline{\underline {108{\rm{ dB}}}} \) (corr. to the nearest integer)
(b) Magnitude of the earthquake
\( = \frac{2}{3}\log (6 \times {10^{15}})  2.9\)
\( = 7.62\) (corr. to 2 d.p.)
Magnitude of the subsequent earthquake
\( = \frac{2}{3}\log (9 \times {10^7})  2.9\)
\( = 2.40\) (corr. to 2 d.p.)
Difference in magnitude \( = 7.62  2.40\)
\( = 5.2\) (corr. to 1 d.p.)
Therefore, the difference in magnitude between these two earthquakes was 5.2 in the Richter scale.