Exponential Inequalities using Logarithms

Exponential Inequalities using Logarithms

Inequalities are worked in exactly the same way except that there is a change of sign when dividing or multiplying both sides of the inequality by a negative number. \begin{array}{|c|c|c|} \hline \log_{2}{3}=1.6>0 & \log_{5}{3}=0.7>0 & \log_{10}{3}=0.5>0 \\ \hline \log_{2}{2}=1>0 & \log_{5}{2}=0.4>0 & \log_{10}{2}=0.3>0 \\ \hline \log_{2}{1}=0 & \log_{5}{1}=0 & \log_{10}{1}=0 \\ \hline \log_{2}{0.5}=-1 \log_{10}{9} \\ [...]
Exponential Growth

Exponential Growth

We will examine situations where quantities are increasing exponentially. This situation is known as $\textit{exponential growth modelling}$, and occur frequently in our real life around us. Population of species, people, bacteria and investment usually $\textit{growth}$ in an exponential way. Growth is exponential when the quantity present is multiplied by a constant for each unit time [...]
Natural Exponential Graphs

Natural Exponential Graphs

$\textit{Natural Exponential Graphs}$ $$y=e^x$$ Natural Exponential Graphs also follow the rule of translations and transformations. Example 1 Sketch the graphs of $y=e^x$ and $y=-e^x$. Show Solution Reflected to the $x$-axis. Example 2 Sketch the graphs of $y=e^x$ and $y=-e^{-x}$. Show Solution Example 3 Sketch the graphs of $y=e^x$ and $y=e^{-x}$. Show Solution Example 4 Sketch [...]
Algebraic Factorisation with Exponents (Indices)

Algebraic Factorisation with Exponents (Indices)

$\textit{Factorisation}$ We first look for $\textit{common factors}$ and then for other forms such as $\textit{perfect squares}$, $\textit{difference of two squares}$, etc. Example 1 Factorise $2^{n+4} + 2^{n+1}$. Show Solution \( \begin{align} \displaystyle &= 2^{n+1} \times 2^{3} + 2^{n+1} \\ &= 2^{n+1}(2^{3} + 1) \\ &= 2^{n+1} \times 9 \\ \end{align} \) Example 2 Factorise $2^{n+3} [...]