## 多维学员专栏 | Additional Maths：Logarithmic and exponential equations发表时间：2020-04-22 13:37 ## Logarithmic and Exponential Functions
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
= ( 且 1)
= ( 且 1)
=
=
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) ：
## 2. GraphsFor , 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
## 3. Laws of LogarithmsThere 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
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
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
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
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: 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.htmlCambridge IGCSE and O level Additional Mathematics CoursebookSpecial Thanks：温昊同学是多维优秀学员， 课程咨询热线：400 9626 006 |