 # Exponential Growth and Decay using Logarithms

Home > Exponential It has been known that how exponential functions can be used to model a variety of growth and decay situations. These included the growth of populations and the decay of radioactive substances. In this lesson we consider more growth and decay problems, focusing particularly on how logarithms can be used in there [...] # 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 Equations using Logarithms

Home > Exponential We can find solutions to simple exponential equations where we could make equal bases and then equate exponents (indices). For example, $2^{x}=8$ can be written as $2^x = 2^3$. Therefore the solution is $x=3$. However, it is not always easy to make the bases the same such as $2^x=5$. In these situations [...] 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 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 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 [...] # Exponential Equations (Indicial Equations) Home > Exponential The equation a^x=y is an example of a general exponent equation (indicial equation) and 2^x = 32 is an example of a more specific exponential equation (indicial equation). To solve one of these equations it is necessary to write both sides of the equation with the same base if the unknown is [...] # 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 [...] 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) 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}} [...] # Rational Exponents (Rational Indices) Home > Exponential \textit{Square Root} Until now, the exponents (indices) have all been integers. In theory, an exponent (index) can be any number. We will confine ourselves to the case of exponents (indices) which are rational number (fractions). The symbol \sqrt{x} means square root of x. It means, find a number that multiply by itself [...] # 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 [...] # Raising a Power to Another Power

Home > Exponential If we are given $(2^3)^4$, that can be written in factor form as $2^3 \times 2^3 \times 2^3 \times 2^3$. We can then simplify using the multiplication using exponents rule as $2^{3+3+3+3} = 2^{12}$. Similarly, if we are given $(5^2)^3$, this means; \( \begin{align} (5^2)^3 &= 5^2 \times 5^2 \times 5^2 \\ [...] 