Problems of growth and decay involve repeated multiplications by a constant number, common ratio. We can thus use geometric sequences to model these situations. $$ \begin{align} \displaystyle\require{color} \color{red}u_{n} &= u_{1} \times r^{n-1} \\\require{color} \color{red}u_{n+1} &= u_{1} \times r^{n}\end{align}$$ Example 1 The initial population of chicken on a farm was $40$. The population increased by $5$% […]

# Tag Archives: Logarithm

# 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 […]

# Logarithm Change of Base Rule

$$\log_{b}{a} = \dfrac{\log_{c}{a}}{\log_{c}{b}}$$ $$\text{for }a,b,c>0 \text{ and } b,c \ne 1$$ For example, \( \begin{align} \log_{3}{8} &= \dfrac{\log_{2}{8}}{\log_{2}{3}} \\ &= \dfrac{\log_{5}{8}}{\log_{5}{3}} \\ &= \dfrac{\log_{10}{8}}{\log_{10}{3}} \\ &\vdots \\ &= 1.8927 \cdots \\ \end{align} \) $\textit{Proof:}$ \( \begin{align} \displaystyle \text{Let } \log_{b}{a} &= x \cdots (1)\\ b^x &= a \\ \log_{c}{b^x} &= \log_{c}{a} &\text{taking logarithm in base […]

# 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 $\textit{logarithms}$ […]

# Logarithmic Equations

We can use the laws of logarithms to write equations in a different form. This can be particularly useful if an unknown appears as an index (exponent). $$2^x=7$$ For the logarithmic function, for every value of $y$, there is only one corresponding value of $x$. $$y=5^x$$ We can, therefore, take the logarithm of both sides […]

# Natural Logarithm Laws

The laws for natural logarithms are the laws for logarithms written in base $e$: $$ \begin{align} \displaystyle \ln{x} + \ln{y} &= \ln{(xy)} \\ \ln{x} – \ln{y} &= \ln{\dfrac{x}{y}} \\ \ln{x^n} &= n\ln{x} \\ \ln{e} &= 1 \\ \end{align} $$ Note that $\ln{x}=\log_{e}{x}$ and $x>0,y>0$. Example 1 Use the laws of logarithms to write $\ln{4} + […]

# Natural Logarithms

After $\pi$, the next weird number is called $e$, for $\textit{exponential}$. It was first discussed by Jacob Bernoulli in 1683. It occurs in problems about compound interest, leds to logarithms, and tells us how variables like radioactivity, temperature, or the human population increase or decrease. In 1614 John Napier knew, from personal experience, that many […]

# Logarithmic Laws

$$ \log_{a}{(xy)} = \log_{a}{x} + \log_{a}{y} $$ $\textit{Proof}$ Let $A=\log_{a}{x}$ and $B=\log_{a}{y}$. Then $a^A = x$ and $a^B=y$. \( \begin{align} a^A \times a^B&= xy \\ a^{A+B} &= xy \\ A+B &= \log_{a}{(xy)} \\ \therefore \log_{a}{x}+\log_{a}{y} &= \log_{a}{(xy)} \\ \end{align} \) $$\log_{a}{\dfrac{x}{y}} = \log_{a}{x} – \log_{a}{y} $$ $\textit{Proof}$ Let $A=\log_{a}{x}$ and $B=\log_{a}{y}$. Then $a^A = x$ […]

# Logarithm Definition

A logarithm determines “$\textit{How many of this number do we multiply to get the number?}$”. The exponent that gives the power to which a base is raised to make a given number.For example, $5^2=25$ indicates that the logarithm of $25$ to the base $5$ is $2$.$$25=5^2 \Leftrightarrow 2=\log_{5}{25}$$If $b=a^x,a \ne 1, a>0$, we say that […]