We started a bit on Logarithm and we learnt how to convert from Index to Log form, and Log to index form.
An example of converting index to log form:
10^x=50 (Index form)
Log1050=x (Log form)
We also learnt the relationship between Log and Indices:
Firstly, Log and Indices are inverse of each other (n^7=x / Lognx=7).
It can be seen from this graph:
To solve Log, sometimes it is easier to convert it to Index form then solve because it is easier.
An example question:
Or in a Log equation:
By Comparing Coefficient,
For the paper this coming Tuesday, remember to work smart and do not get stuck at one question as time is a factor in the paper. DO NOT over-read the question and over-answer the question as well.
For questions that have the word 'Hence', you MUST use the solution from the previous part of the same question. 'Otherwise' means you can either resolve the question or use the solution from the previous part of the same question.
Plotting of Quadratic Graphs (Revision):
General form of Quadratic: ax^2+bx+c
Take a look at this graph:
At the blue line (x^2), there is no coefficient.
-When the coefficient is bigger than 1 it will make the graph narrower, and if the coefficient is less than 1 it will make the graph wider. When the coefficient is negative, it will invert the graph.
The two x intercepts are 4 and -2, and the y intercept is -8.
Sample Questions for Quadratic Graphs:
What is given in the question?
1. Table of values (DON'T spend too much time trying to draw a nice nice table)
What must you show?
1. Crosses on points
4. Labeling of equation
1. Line of symmetry (Turning point)
2. Gradient of Tangent (Tangent at x=1) ( Use the formula (y1-y2)/(x1-x2) )
3. Plotting of the graph
4. Find the value of x when y = ____.
5. Plotting of a linear graph and find the intersection.
Because of the late posting (Sorry), I will post up a clearer summary of Surds:
Surds are bascially numbers left in 'square root form' or 'cube root form', etcetc, and are irrational. We leave them it this form because writing them out in decimal form would not be a very good way of expressing.
Firstly, there are a few operations and rules that we must know of, just like in Indices.
Operations of Surds:
1. √a x b = b√a
2. √a (divide) b = √a (over) b
Rules: (When a rational number is multiplied or divided by a surd, the result will be a surd.)
3. √a x √b = √ab (√3 x √4 = √12)
4. √a x √a = √a^2 = a (√8 x √8 = 8)
5. √a (divide) √b = √a (over) √b = √a/b (√20 / √4 = √5)
NOTE: √(a+b) ≠ √a + √b and √(a-b) ≠ √a - √b
6. x√a + z√a = (x+z)√a (IT'S NOT (x+z) ROOT OF A) (5√2 + 3√2 = 8√2)
7. x√a - z√a = (x-z)√a (IT'S NOT (x-z) ROOT OF A) (5√2 - 3√2 = 2√2)
Rationalizing of Surds:
√a + √b and √a - √b are conjugate surds. This means that the product of them is a rational number.
(√a + √b)(√a - √b) = (√a)^2 - (√b)^2
Note: a-b is a rational number, thus we can use conjugate surds to rationalize the denominator.
Solving equations in Surds:
1-Equations for surds can be solved by squaring both sides, to remove the square root.
2-Sometimes we have to square both sides TWICE when the equation involves sum or difference of 2 or more surds.
Once again sorry for the late posting .