The following examples show that you do not need to apply the quotient rule for differentiating when the denominator is a constant. Please see the following cases with the same question. 1. $\textit{Application of the quotient rule}$ \( \begin{align} \displaystyle \dfrac{d}{dx}\dfrac{48x – 4x^3}{3} &= \frac{{\frac{d}{{dx}}\left( {48x – 4{x^3}} \right) \times 3 – \left( {48x – […]

# Author Archives: iitutor

# Logarithmic Equations

Home > iitutor 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 [...]

# Natural Logarithm Laws

Home > iitutor 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 [...]

# Natural Logarithms

Home > iitutor 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 [...]

# Logarithmic Laws

Home > iitutor $$ \log_{a}{(xy)} = \log_{a}{x} + \log_{a}{y} $$ $\textit{Proof}$ 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 [...]

# Logarithm Definition

Home > iitutor 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 [...]

# Natural Exponential

Home > iitutor 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 [...]

# Logarithms in Base 10

Home > iitutor Many positive numbers can be easily written in the form $10^x$. $$ \begin{align} 10\ 000 &= 10^4 \\ 1000 &= 10^3 \\ 100 &= 10^2 \\ 10 &= 10^1 \\ 1 &= 10^0 \\ 0.1 &= 10^{-1} \\ 0.01 &= 10^{-2} \\ 0.001 &= 10^{-3} \\ \end{align}$$ All positive numbers can be [...]

# Exponential Decay

Home > iitutor 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 > iitutor 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 > iitutor $\textit{Natural Exponential Graphs}$ $$y=e^x$$ 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 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)

Home > iitutor 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 > iitutor $\textit{Factorisation}$ 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)

Home > iitutor $\textit{Algebraic Expansion with Exponents}$ 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 > iitutor 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 > iitutor $\textit{Square Root}$ 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 > iitutor 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 > iitutor 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 \\ [...]

# Perfect Numbers

Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers with a bizarre property that they can be formed by adding up all the smaller numbers that make up their divisors. The number $6$ is the first perfect number, because it may be divided by […]