Functions

5.4 Solving Exponential and Logarithmic Equations

Fundamental of Solving Exponential and Logarithm Functions

In solving equations, the most important identity of both exponential and logarithmic functions are that they are one-to-one functions. In other words,

$a^{x}=a^{y}$ or $\log_{a}{x}=\log_{a}{y}$ implies that $x=y$ .

Example

3^{x-1}=81=3^{4}

Basic Trigonometric Equations (Sine and Cosine)

The most basic equations in polynomials would be linear equations, such as $ax+b=c$ (which gives you $x=\frac{c-b}{a}$ ). The trigonometric equivalent ones would be $\sin t =a$ or $\cos t =b$ , and you are supposed to solve for $t$ .

Be careful, as solving trigonometric equations are not as simple. Refer to the examples below.

For each such equations, unless a restriction on $x$ (or more commonly used in trigonometric, $\theta$ ), there will be infinite or no solutions for $x$ . This is because of the periodic and symmetric properties that they have.

Solving Equations

Every now and then, you will be challenged with questions to solve .

Suppose we have $y=f(x)$ and $y=g(x)$ as two separate functions. If $f(x)=g(x)$ , it means that the graph of both functions intersect. Note that this tells nothing about the number of possible values for $x$ .

Simultaneous Equations: Polynomials

We will first look at how this type of questions looks like, with polynomials first.

The steps are rather simple:

Application of Functions

Functions is a very useful tool in expressing relations, even in our daily life. Below are a few examples.

Composite Functions

A composite function is different from sum or product of functions; it is like a function acting on the result of another. If you picture functions as machines (with $x$ as the input, and $f(x)$ as the output), it is something like the diagram shown.

Note: Here, $f(x)=3 x+2,\ g(x)=x^{2}$ , and $x=3$ .

In comparison, $(f+g)(x)$ is more like tow machines working separately with the same input $x$ , and their two, separate outputs are placed together and packaged.

Just like how the processes of manufacturing has an order, so does composite functions. It is important which functions comes first, and which comes next. Using the example above where $f$ is the first function), we write the composition of $g$ with $f$ (say we call it as $h$ ) as $h=g \circ f$ (read as ‘composition of $f$ followed by $g$ ‘). The rule of $h(x)$ is given by

3.4 Trigonometric Function Laws

Note: These are already covered in the notes 1.5 Trigonometric Functions Background Knowledge, when trigonometric functions are first introduced. Here, it will be simply a reiteration of the formulas.

Trigonometric Function Laws

Sine, cosine, tangent can be understood as follows:

$\sin (\theta)=\frac{\text { opposite }}{\text { hypotenuse }}$

$\cos (\theta)=\frac{\text { adjacent }}{\text { hypotenuse }}$

$\tan (\theta)=\frac{\text { opposite }}{\text { adjacent }}$

and in particular we have

The Logarithm Function’s Algebraic Properties

There are four properties in logarithms to consider:

i) $\log_{a}{mn}=\log_{a}{m}=\log_{a}{n}$

ii) $log_{a}{(\frac{m}{n})}=\log_{a}{m}-\log_{a}{n}$

iii) $log_{a}{(m^{p})}=p\cdot{\log_{a}{m}}$

In particular when $p=-1$ , we have $log_{a}{(\frac{1}{m})}=-\log_{a}{m}$

The Exponential Function’s Algebraic Properties

There are six properties in exponentials to consider:

i) $a^{x} \times a^{y} = a^{x+y}$

ii) $a^{x} {\div} a^{y} = a^{x-y}$

iii) $(a^{x})^{y}=a^{xy}$

3.1 Introduction to Functional Relations

Function Notation and Identities

Many of the properties which have been investigated for the functions introduced in the previous chapters may be expressed using function notation.

Below includes a few examples:

$\ln x+\ln y=\ln (x y)$ becomes $f(x)+f(y)=f(x y)$ where $f(k)=\ln k$

$\ln x-\ln y=\ln \left(\frac{x}{y}\right)$ becomes $f(x)-f(y)=f\left(\frac{x}{y}\right)$ where $f(k)=\ln k$

Remainder Theorem and Factor Theorem

Previously in 2.1 Polynomial Function Introduction and Algebraic Operations we have seen that we can write division of polynomials in the form $P(x)=D(x) Q(x)+R(x)$ .

Now we are going to set $D(x)=x-a$ . Therefore, as $\operatorname{deg}(R)<\operatorname{deg}(D)$ , $R(x)$ must be a constant, which we denote as $R$ .

Rewriting the equation gives

$P(x)=(x-a) Q(x)+R$

Simply set $x=a$ gives $P(x)=(a-a) Q(x)+R=0+R=R$ . Thus, $R=P(a)$ .

Now we generalize $D(x)$ into any linear polynomial, i.e. $D(x)=a x+b$ , where $a,\ b$ are integers. Therefore, now $P(x)=(a x+b) Q(x)+R$ and substitute $x=-\frac{b}{a}$ (to make $D(x)=0$ ) gives

Functions

5.4 Solving Exponential and Logarithmic Equations

Fundamental of Solving Exponential and Logarithm Functions

5.3 Solution of Trigonometric Equations

Basic Trigonometric Equations (Sine and Cosine)

5.2 Simultaneous Equations: Polynomials

Solving Equations

Simultaneous Equations: Polynomials

5.1 Application of Functions

Application of Functions

4.1 Composite Functions