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多维学员专栏 | Additional Maths:Logarithmic and exponential equations

发表时间:2020-04-22 13:37

多维从成立至今,一直秉承着:“用最优秀的人培养更优秀的人”的理念。
多维的老师厉害,学生更厉害,而且我们的学员可不仅仅只有傲人的成绩,更有独具一格的创造力。
为了展现多维学员的创造力,我们开设了【多维学员专栏】今天专栏的分享来自多维优秀学生——温昊同学的原创文章。
温昊同学现在就读于城市绿洲,今天给大家分享的是「对数函数和指数函数」,让我们来看看吧~
以下正文:

Logarithmic and Exponential Functions

"The book of nature is written in the language of Mathematic"---Galileo

Logarithmic and exponential functions is the 6th chapter of the Additional Mathematics. This post will discuss several identities about logarithms and some exam-style questions.

Content

  1. Definition
  2. Graphs
  3. Rules of calculation
  4. Examination questions

1. Definition

Exponential Function (指数函数):

= (1)

Logarithmic Function (对数函数):

= (1)


Common Logarithm (常用对数) :

=

Natural Logarithm (自然对数) :

=


Euler's number

is one of the most important numbers in mathematics. It can be found in a lot of areas and has a wide range of fascinating properties, which we will discuss later in the article.

is an irrational number and the first few digits are: = 2.718281828459...

We can define in 3 ways:

  • Using Limit (极限定义)

=

  • Using factorial (阶乘定义):

= +++++ ...

  • Using Euler's Identity (欧拉恒等式) :


= imginary number


2. Graphs

For , the domain is , range is

For , the domain is , range is   ,

  • must pass through the point (0,1) because anything to the power of equals to 1

  • must pass through the point (1,0) becuase it is an inverse function of

For 0<<1, it is a decresing function
When a>1, it is an increasing function
For , when 0<<1, it is a decreasing function
When , it is an increasing function
is greater than 1, so it will be an increasing function

3. Laws of Logarithms

There are several laws we have to remember.

Once you understand these laws, it should be pretty easy to do these questions. But please remember, these laws only apply if and are both positive and and 1.

Here are some important laws you must memorize:

  • Multiplication:


  • Division :   


  • Power:


  • Change of base:


if , then

  • Other useful laws:

4. Examination Questions

  1. Solve the following simultaneous euqations.

Solution:

First, using the rule

,

we can express 2+ into 2.

(Note: can be written as )

Then applying the product rule, this expression can be changed to .

By obeserving both sides, we know .

The second euqation tells us . So now, we have two simplified equations:

It is very simple now when we reach this point. After solving these two simultaneous equations, we can calculate the value of and

From Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q3 Nov 2014

  1. Using the sbstitution , solve

Solution:

First, let's express into and can be written as , therefore the left hand side can be rewritten as .

Then, we can substitute into this formula and we will have .

Now, all we need to do is to solve this quardratic equation .

This euqation can be factorised =0, so u =1 or .

Replacing with , we can calculate x or


Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q4 Nov 2012

  1. Solve the following equation:

Solution:

First, we will need to change the base of into .

Then, we can simplify the right side to 3+2.(Note: 3 can be rewrite as 3)

Using the multification law, 3+2 can be written as .

Comparing both sides, we can know .

Now it become very easy to solve this quardratic equation, can be factorised to or

  1. Sketch the graph of

Solution:

I believe many of the students can do the questions above very easily but they are struggling about drawing graphs.

The very first thing we need to do is to find out the intercepts.

When

y intercept is (0,-1)

When

x intercept is (-0.07,0)

After finding the intercepts, we need to consider the limits.

As so y

Hence the asymptote is -5

As

Using the information above, we can sketch something like this (p.s. this is a very ugly sketch that i've drawn lol):

Here is the graph DESMOS create:

Powered by DESMOS

好了,这次的内容到这就结束了。总而言之,对数函数和指数函数是Additional Maths里中等偏上难度的一章。只要掌握了基本的运算法则和清晰的认识它图像的性质,所有题目应该都不在话下。

参考资料:
https://www.mathsisfun.com/numbers/e-eulers-number.html
Cambridge IGCSE and O level Additional Mathematics Coursebook
Special Thanks:
张艺炜老师 for supervising the work
多维教育 for massive supports

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