### Exponential and logarithmic functions - (3.4-3.7)

#### Understand the properties of exponential functions and logarithmic functions and recognise the features of their graphs

• $$f(x)=log_{a}x$$

• $${\rm{y}} = {\log _{\rm{a}}}x$$
- 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
• $${\rm{y}} = {{\rm{a}}^{\rm{x}}}$$
- 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

• Solve 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) $$\log (x + 5) - \log (2x - 1) = \log 6$$
• $$\begin{array}{1}(a)\qquad{6^{x{\rm{ }} + {\rm{ }}2}} = {3^{3x{\rm{ }} - {\rm{ }}4}}\\\log {6^{x{\rm{ }} + {\rm{ }}2}} = \log {3^{3x{\rm{ }} - {\rm{ }}4}}\\(x + 2)\log 6 = (3x - 4)\log 3\\x\log 6 + 2\log 6 = 3x\log 3 - 4\log 3\\3x\log 3 - x\log 6 = 2\log 6 + 4\log 3\\x = \frac{{2\log 6 + 4\log 3}}{{3\log 3 - \log 6}}\end{array}$$
$$= \underline{\underline {5.30}}$$ (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 real-life situations

• - 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.
• (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

(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.)
• (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.