Exponentials
Exponential Growth and Decay using Logarithms
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 solution. Population Growth […]
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<0 & \log_{5}{0.5}=-0.4<0 […]
Exponential Equations using Logarithms
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, we use $\textit{logarithms}$ […]
Natural Exponential
We learnt that the simplest exponential functions are of the form $y=a^x$ where $a>0$, $a \ne 1$. We can see that for all positive values of the base $a$, the graph is always positive, that is $a^x > 0$ for all $a>0$. There are an infinite number of possible choices for the base number. However, […]
Exponential Decay
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 &= 30 \\ R_1 […]
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. The 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
$$y=e^x$$ Natural Exponential Graphs also follow the rule of translations and transformations. Example 1 Sketch the graphs of $y=e^x$ and $y=-e^x$. Reflected to the $x$-axis. Example 2 Sketch the graphs of $y=e^x$ and $y=-e^{-x}$. Example 3 Sketch the graphs of $y=e^x$ and $y=e^{-x}$. Example 4 Sketch the graphs of $y=e^x$ and $y=e^{x+1}$. Translated to left. […]
Exponential Graphs
Functions of the form exponential, where the base is a positive real number other than 1 are called exponential graphs or exponential functions.
Exponential Equations (Indicial Equations)
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 an exponent (index) […]
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