1.0 Manipulating Surds

Surds are that cannot be expressed as a fraction of two integers. They are represented by the form

irrational numbers

Real numbers that can't be expressed in terms of simple fractions

$\sqrt{x}$

(or $x^{\frac{1}{2}}$

in index form) and can be manipulated in the following ways:Note 1.0.0 - Surd manipulation

1) Multiplying surds: $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$

2) Multiplying similar surds:

$\sqrt{a}\times\sqrt{a}=a$

3) Dividing surds:

$\frac{\sqrt{b}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

4) Rationalising surds:

$\frac{a}{\sqrt{b}+\sqrt{c}}\times\frac{a\left( \sqrt{b}-\sqrt{c} \right)}{\sqrt{b}-\sqrt{c}}=\frac{a\sqrt{b}-a\sqrt{c}}{b-c}$

Surds are generally used in precision applications like reducing noise to detect patterns in signals in computer science, all they way to our current understanding of the underlying mathematics governing quantum mechanics (such as the logic behind the Hadamard gate). Some examples to get you started are presented below, and a series of questions follow that.

Example 1.0.0 - Basic surd simplification and rationalization

1) Simplify $\sqrt{12}$

2) Simplify

$\frac{\sqrt{20}}{2}$

3) Rationalise

$\frac{2}{\sqrt{6}-2}$

giving your answer in the form $a+b\sqrt{6}$

1) When simplifying surds, we need to see what can be factored out of the number under the square root. In this case, $\sqrt{12}$

. We can see that 12 can be factored into 4 and 3, and since 4 is a perfect square, we can factor it out of the square root (as 2). This gives us $\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}$

.2) When simplifying fractions with surds, we need to

rationalise

The process of changing a fraction's denominator from an irrational number to one that's rational

$\frac{2\sqrt{5}}{2}=\sqrt{5}$

.3) When rationalizing, we need to multiply the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of

$\sqrt{6}-2$

is $\sqrt{6}+2$

, so we multiply the numerator and denominator by this conjugate. This gives us $\frac{2\left( \sqrt{6}+2 \right)}{6-4}=\frac{2\sqrt{6}+4}{2}=2+\sqrt{6}$

.The examples above were just simple representations, but they provide the axioms for which you can solve more complex surd problems. The questions below will test your understanding of basic surd manipulation, and further questions involving rationalization. Try to do these without a calculator!

Problem 1.0.0 - Surd manipulation and rationalization

1) Simplify $\sqrt{18}$

2) Simplify

$\frac{\sqrt{45}}{3}$

3) Simplify

$2\sqrt{3}\times 5\sqrt{6}$

4) Simplify

$\frac{\sqrt{12}}{\sqrt{3}}$

5) Simplify

$\frac{\sqrt{15}}{\sqrt{5}}$

6) Simplify

$\sqrt{50}-\sqrt{18}$

, giving your answer in the form $a\sqrt{2}$

7) Simplify

$\sqrt{500}+\sqrt{125}$

, giving your answer in the form $a\sqrt{5}$

8) Simplify

$2\sqrt{98}+\sqrt{50}$

9) Simplify

$9\sqrt{72}+6\sqrt{8}$

10) Rationalise

$\frac{5x}{\sqrt{2}-2x}$

Problem 1.0.1 - Further rationalization

1) Rationalise $\frac{3x^2-4x}{\sqrt{x}+2}$

2) Express

$\frac{2+\sqrt{7}}{\sqrt{7}-2}$

in the form $a+b\sqrt{7}$

3) Express

$\frac{26}{4-\sqrt{3}}$

in the form $a+b\sqrt{3}$

4) Show that

$\frac{2\sqrt{18}-2}{\sqrt{2}-1}$

can be written in the form $a+b\sqrt{2}$

5) Rationalise

$\frac{\sqrt{5}+10}{5-2\sqrt{5}}$

, giving your answer in the form $a+b\sqrt{5}$

6) Show that

$\frac{(\sqrt{8}+\sqrt{2})^2}{3\sqrt{2}-4}$

can be written in the form $a+b\sqrt{2}$

7) Show that

$\frac{3-\sqrt{125}}{(2+\sqrt{5})^2}$

can be written in the form $a+b\sqrt{5}$

8) Show that

$\frac{(\sqrt{6}+6)^2}{3+\sqrt{6}}$

can be written in the form $a+b\sqrt{6}$

9) Solve

$\sqrt{75}z+12=\sqrt{3}z$

, giving you answer in the form $a\sqrt{3}$

10) Solve

$3x=\sqrt{6}x+9$

, giving your answer in the form $a+b\sqrt{6}$

1.1 Graphs of Surd Functions

When trying to solve surds involving unknown variables (e.g. when trying to find the x-intercept of a surd graph), you might have noticed that it's very hard to get whole (or close to whole) number answers. This further proves how you're not able to express surds as simple fractions! If we look at the example below of the function

$\frac{\sqrt{7x}}{2}-3$

, we can find that the x-values end up being large decimals. Graph 1.1.0 - Graph of

$\frac{\sqrt{7x}}{2}-3$

The graph of a surd function

$\frac{\sqrt{7x}}{2}-3$

is a curve that starts at the origin and moves in a positive direction along the x-axis, with an initial but sudden increase of y values near the origin. We can apply the shape of this graph in real life to the velocity of a pendulum. As you let go of the pendulum at a maximum point, its velocity is sudden and large, but as it gets closer to the point of equilibrium, its velocity slowes down (hence the smoother gradient change in later x values).1.2 Introduction to Logarithms

Logarithms are the inverse of exponentials, and are used to solve equations where the variable is in the exponent. The are used when dealing with real-life problems like half-life decay graphs for radioisotopes, the decibel scale, the pH scale, and the Richter scale. They are generally expressed in the form

$\log_xa$

, but can be expressed in terms of $e$

(which I will cover later in this chapter). Below are the general rules involving log conversions:Note 1.2.0 - Converting to log form

1) General log: $N=a^x \rightarrow \log_aN=x$

2) Common log:

$N=10^x \rightarrow \log N=x$

3) Natural log:

$N=e^x \rightarrow \ln N=x$

You may have noticed, there are several ways to convert TO a logarithm. Remember, a logarithm is the inverse of an exponential, and if we try to solve it using regular algebra rules, we find ourselfs unable to. This is where converting to log form can help especially when finding variables WITHIN exponents. Below are some examples to get you started, followed by some simple conversion questions.

Example 1.2.0 - Basic log conversion

1) Convert $2^5=32$

to log form2) Convert

$10^2=100$

to log form3) Convert

$e^3=20$

to log form1) To convert $2^5=32$

to log form, we need to see what the base is. In this case, the base is 2, and the exponent is 5. This gives us $\log_232=5$

2) To convert

$10^2=100$

to log form, we need to see what the base is. In this case, the base is 10, and the exponent is 2. This gives us $\log100=2$

3) To convert

$e^3=20$

to log form, we need to see what the base is. In this case, the base is $e$

, and the exponent is 3. This gives us $\ln20=3$

Above show simple conversions, but now we need to test them out in functions. Below are some examples.

Problem 1.2.0 - Log usage

1) Solve $\log_3x=4$

2) Solve

$\log_x243=5$

3) Solve

$\ln x=2$

4) Solve

$\log_2x=3$

5) Solve

$\log_5x=1$

6) Solve

$\ln x=3$

7) Solve

$\log_4x=2$

8) Solve

$\log_7x=3$

9) Solve

$\ln x=4$

10) Solve

$\log_6x=2$

Now that you're familiar with basic log conversion, here's some formulae you'll need to memorize to be able to solve more complex log equations.

Note 1.2.1 - Log rules

1) Multiplying logs: $\log_aN+\log_aM=\log_aNM$

2) Dividing logs:

$\log_aN-\log_aM=\log_a\frac{N}{M}$

3) Power conversion:

$\log_aN^m=m\log_aN$

4) Base conversion:

$\log_aN=\frac{\log_bN}{\log_ba}$

5) Equality: when

$\log_bn=\log_bm$

, then $n=m$

Below are some examples of the application of these rules. For now, you wont need to know the 4th conversion, as it will be covered in the next section.

Example 1.2.1 - Solving equations involving logs

1) Solve $2\log_{10}x=1+\log_{10}5-\log_{10}2$

2) Solve

$\log_6x+\log_6(x+5)=2$

, where $x>0$

3) Solve

$\log_3(2x-1)=2$

1) To solve $2\log_{10}x=1+\log_{10}5-\log_{10}2$

, we first move all the logs to one side, and then use both the multiplication and division rule like so: multiply $\log_{10}x^2$

and $\log_{10}2$

and put it all over 5. This gives us $\log_{10} \left( \frac{2x^2}{5} \right)=1$

. We can then remove the log, and solve like a regular equation: $\frac{2x^2}{5}=10$

, then $x^2=25$

, and finally $x=5$

.2) To solve

$\log_6x+\log_6(x+5)=2$

, we can use the multiplication rule to combine the logs into one. This gives us $\log_6(x(x+5))=2$

, and we can then remove the log to get $x(x+5)=36$

, which gives us $x=4$

(you can either complete the square, or use the quadratic formula to get this answer).3) To solve

$\log_3(2x-1)=2$

, we can remove the log to get $2x-1=9$

, and then solve for x to get $x=5$

.Now that you've seen some examples, here are some problems for you to try out.

Problem 1.2.1 - Solving and manipulating logs

1) Solve $\log_4x+\log_4(x+1)=1$

2) Given that

$2\log_4(3x+5)=\log_4(3x+8)+1,x>-\frac{5}{3}$

, show that $9x^2+ax+b=0$

where $a$

and $b$

are constants to be found3) Solve

$7^{x+2}=3$

, giving your answer in the form $x=\log_7a$

where $a$

is a rational number in its simplest form4) Solve

$\log_5(3-2x)+\log_5(2+x)=1$

where $x$

is a real number5) Express

$2\log_3x-\log_3(x+4)$

as a single logarithm6) Find any real values of

$x$

such that $2\log_4(2-x)-\log_4(x+5)=1$

7) Solve the equation

$\log_2x+2\log_23=\log_2(x+5)$

8) Given that

$3\log_3(2x-1)=2+\log_3(14x-25)$

, show that $ax^3+bx^2+cx+d=0$

where $a, b, c$

and $d$

are integers to be found9) Find the exact solution of

$\log_4(3y+2)=\log_4(y+2)+\log_4(2y-3)$

10) Solve

$2\log_2(x+1)=\log_2(x+13)+1$

1.3 Further Logarithms

Now that you know the basic rules of log manipulation and can solve them, we need to understand how base changes work and how they can help us. The questions above all had the same base, which makes them easy to solve, but there are log functions that require base changes.

Example 1.3.0 - Base changes

1) Changing bases: $\log_aN=\frac{\log_bN}{\log_ba}$

If we look above, we see that another base 'b' is created. We can derive this from other log rules as followed:

1) Let: $\log_aN=x$

2) Convert to exponential form:

$N=a^x$

3) Take the log (

$\log_b$

) of the new bases: $\log_bN=\log_ba^x$

4) Use the power rule:

$\log_bN=x\log_ba$

5) Divide by

$\log_ba$

: $\log_aN=\frac{\log_bN}{\log_ba}$

Now lets take a look at some examples. These are all completed using base change rules, and when completing the questions below, you should also use base changes!

Example 1.3.1 - Simplifying equations using base changes

1) Find the value of $\log_918$

2) Find the value of

$x$

in $3^x=23$

3) Given that

$\log_23=\log_43^x$

, find the value of $x$

1) To find the value of

$\log_918$

, we need to change the base. We can change it to use base 10 (remember $\log_{10}x=\log x$

) which gives us $\frac{\log 18}{\log 9}=1.32$

.2) We first need to convert to exponential form which would give

$\log_323=x$

. From there, we apply the base change rule to get $\frac{\log 23}{\log 3}=2.85$

.3) We don't need to use the power rule at the start here, so we will focus yet again on changing the base. We will apply this rule to both sides to get

$\frac{\log_23}{\log_22}=\frac{\log_23^x}{\log_24}$

. We then simplify the logs which gets us $\log_23=\frac{\log_23^x}{2}$

. We can finally use the power rule here to show that $2\log_23=\log_23^x \therefore x=2$

The examples above show how base changes can be used to simplify log equations. The questions below will test your understanding of base changes and how they can be used to simplify log equations. They won't be hard because you won't find many questions about base changes.

Problem 1.3.0 - Further log questions involving base changes

1) Find the value of $\log_512$

2) Find the value of

$x$

in $2^x=32$

3) Given that

$\log_32=\log_22^x$

, find the value of $x$

4) Find the value of

$\log_2\frac{1}{8}$

5) Find the value of

$x$

in $5^x=125$

6) Given that

$\log_125=\log_55^x$

, find the value of $x$

7) Find the value of

$\log_3\frac{1}{27}$

8) Find the value of

$x$

in $4^x=64$

9) Given that

$\log_64=\log_44^x$

, find the value of $x$

10) Find the value of

$\log_4\frac{1}{64}$

1.4 Eulers Number

Eulers number

$e$

is a mathematical constant (just like $\pi$

) which can be used to explain things from population models all the way to radioactive decay. As you saw earlier, it has the definition $N=e^x \rightarrow \ln N=x$

. Below are some examples to get you started!Example 1.4.0 - Solving equations with

1) Find the exact solution, in their simplest form, to the equation $e$

$e^{3x-9}=8$

2) Solve for

$x$

: $5e^{4x-3}=11$

, giving your solution in exact form3) Solve for

$x$

: $\ln (3x-7)+5=8$

, giving your solution in exact form1) We first have to convert the equation to a logarithmic form, which gives us $\ln 8=3x-9$

. We can then solve for $x$

to get $x=\frac{9+\ln 8}{3}$

. This can be simplified to $\ln 2+3$

.2) We first have to move the constant to the other side so we're able to log the equation:

$e^{4x-3}=\frac{11}{5}$

, and then $4x-3=\ln \left( \frac{11}{5}\right)$

. Then, same as always, we just rearrange to get the exact value: $4x=\ln \left( \frac{11}{5}\right)+3$

, which results in: $x=\frac{\ln \left( \frac{11}{5}\right)+3}{4}$

.3) As per usual, we have to rearrange the equation to get

$\ln (3x-7)=3$

, and then $3x-7=e^3$

, and finally $x=\frac{e^3+7}{3}$

.There's not much else for now, so here's a few questions on solving logs with eulers number:

Problem 1.4.0 - Solving equations with

1) The curve C has parametric equations $e$

$x=\ln t$

and $y=t^2-2$

when $t>0$

. Find the cartesian equation of C2) The value of a car can be calculated from the formula

$V=17000e^{-0.25t}+2000e^{-0.5t}+500$

where $V$

is the value of the car in dollars and $t$

is the age in years. Calculate the exact value of $t$

when $V=9500$

3) The population

$p$

of a species at time $t$

years is mapped by the equation $p=\frac{2800ae^{0.2t}}{1+ae^{0.2t}}$

where $a$

is a constant. Use the equation with $a=0.12$

to predict the number of years before the population of the species reaches 18504) A curve with the equation

$(8-x)\ln x$

(where $x>0$

) cuts the x-axis at two points. What are the $x$

-coordinates of these points?5) Without using a calculator, solve an equation to find the exact

$x$

-coordinate of the intersection points of the functions $y=e^{2x}$

and $y=8e^{-x}$

6) Find the exact solutions of

$e^x+3e^{-x}=4$

7) Find algebraically the exact solutions of

$\ln (4-2x)+\ln (9-3x)=2\ln (x+1)$

where $-1<x<2$

8) A curve has the equation

$y=3\ln (2x-a)$

, where $a$

is a positive constant. Sketch the curve and determine the $x$

-coordinate of $P$

in terms of $a$

9) Find the exact solution, in its simplest form, to the equation

$3^xe^{4x}=e^7$

10) The point

$P$

lies on the curve $y=4e^{2x+1}$

. With $P$

having a $y$

-coordinate of 8, find, in terms of $\ln 2$

, the $x$

-coordinate of $P$

Well done for completing this chapter! You can either have a go at the ECQs below, or head on to the next chapter.

1.ECQ End of Chapter Questions

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