# 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 [...]

# Natural Exponential

Home > 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 [...]

# 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 \\ [...]

# Zero Index

Home > Exponential Any base that has a power of zero has a value of one. it does not matter whether the base is a number or a pronumeral. If the power of zero; its value is one. We can show this by looking at the following example which can be simplified using two different [...]

# Division using Exponents (Indices)

Home > Exponential If we are given $a^8 \div a^3$, we can also write this as $\dfrac{a^8}{a^3}$, which means $\dfrac{a \times a \times a \times a \times a \times a \times a \times a}{a \times a \times a}$. As there are $8$ factors of $a$ on the top line (numerator), and $3$ factors of $a$ [...]