Exponential Inequalities using Logarithms

Exponential Inequalities using Logarithms

Home > Exponential 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 [...]
Exponential Decay

Exponential Decay

Home > Exponential Consider a radioactive substance with original weight $30$ grams. It $\textit{decays}$ or reduces by $4\%$ each year. The multiplier for this is $96\%$ or $0.96$. When the multiplier is less than $1$, we call it as $\textit{Exponential Decay}$. If $R_n$ is the weight after $n$ years, then: \( \begin{align} \displaystyle R_0 &= [...]
Exponential Growth

Exponential Growth

Home > Exponential 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 [...]
Natural Exponential Graphs

Natural Exponential Graphs

Home > Exponential $\textit{Natural Exponential Graphs}$ $$y=e^x$$ Natural Exponential Graphs 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 [...]
Algebraic Factorisation with Exponents (Indices)

Algebraic Factorisation with Exponents (Indices)

Home > Exponential $\textit{Factorisation}$ 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 [...]
Algebraic Expansion with Exponents (Indices)

Algebraic Expansion with Exponents (Indices)

Home > Exponential $\textit{Algebraic Expansion with Exponents}$ Expansion of algebraic expressions like $x^{\frac{1}{3}}(4x^{\frac{4}{5}} - 3x^{\frac{3}{2}})$, $(4x^5 + 6)(5^x - 7)$ and $(4^x + 7)^2$ are handled in the same way, using the same expansion laws to simplify expressions containing exponents: $$ \begin{align} \displaystyle a(a+b) &= ab+ac \\ (a+b)(c+d) &= ac+ad+bc+bd \\ (a+b)(a-b) &= a^2 -b^2 [...]
Complicated Exponent Laws (Index Laws)

Complicated Exponent Laws (Index Laws)

Home > Exponential So far we have considered situations where one particular exponents law was used for simplifying expressions with exponents (indices). However, in most practical situations more than one law is needed to simplify the expression. The following example shows simplification of expressions with exponents (indices), using several exponent laws. Example 1 Write $64^{\frac{2}{3}}$ [...]
Negative Exponents (Negative Indices)

Negative Exponents (Negative Indices)

Home > Exponential Consider the following division: $$\dfrac{3^2}{3^3} = 3^{2-3} = 3^{-1}$$ Now, if we attempt to calculate the value of this division: $$\dfrac{3^2}{3^3} = \dfrac{9}{27} = \dfrac{1}{3}$$ From this conclusion we can say that $3^{-1} = \dfrac{1}{3}$. This conclusion can be generalised: $$a^{-1} = \dfrac{1}{a}$$ Example 1 Write $4^{-1}$ in fractional form. Show Solution [...]
Laws of Exponents (Index Laws)

Laws of Exponents (Index Laws)

$\textbf{Laws of Exponents (Index Laws)}$ $a^x \times a^y = a^{x+y}$ To $\textit{multiply}$ numbers with the $\textit{same base}$, keep the base and $\textit{add}$ the exponents. $\dfrac{a^x}{a^y} = a^x \div a^y = a^{x-y}$ To $\textit{divide}$ numbers with the $\textit{same base}$, keep the base and $\textit{substract}$ the exponents. $(a^x)^y = a^{x \times y}$ When $\textit{raising a power to […]

Index Notation

Index Notation

Home > Exponential $\textbf{Index Notation}$ A convenient way to write a product of $\textit{identical factors}$ is to use $\textbf{exponential}$ or $\textbf{index notation}$. Rather than writing $5 \times 5 \times 5 \times 5$, we can write this product as $5^4$. The small $4$ is called the $\textbf{exponent}$ or $\textbf{index}$, and the $5$ is called the $\textbf{base}$. [...]