Course

O Level Additional Mathematics: Logs and Coordinate Geometry (4037)

Name: O Level Additional Mathematics: Logs and Coordinate Geometry (4037)
Price: 1299 BDT

Logarithmic and exponential functions, straight-line graphs and the coordinate geometry of the circle for Cambridge O Level Additional Mathematics 4037, taught step by step with worked examples, practice and exam questions.

44 lessons8h 12mEnglish

What you'll learn

Read a logarithm as a power and move between logarithm and index form
Use the laws of logarithms, including change of base, to simplify and solve equations
Solve exponential equations of the form a^x = b, including ones that become quadratics
Sketch y = ke^(nx) + a and y = k ln(ax + b), state their asymptotes and use e^x and ln x as inverses
Find the length, midpoint, gradient and equation of a line, including a perpendicular bisector
Transform curved relationships to straight-line form and find the unknown constants
Find the centre and radius of a circle from its equation in either form
Find where a line meets a circle and decide whether it is a tangent, a chord or a miss
Find equations of tangents, and decide how two circles meet and find their common chord

Requirements

Cambridge O Level Mathematics 4024 or equivalent, including indices, gradients and Pythagoras
Solving quadratic equations and completing the square

Course content

44 lessons · 8h 12m

6.1-6.3 · Logarithmic and exponential functions

Read a logarithm as a power, use the laws of logarithms and change of base, solve exponential equations, and sketch and invert the graphs of e^x and ln x.

0/16

Logarithms to base 10

lg 1000 = 3 because 10^3 = 1000. A logarithm to base 10 is the power you put on 10 to get the number.

Preview10m 2.

Logarithms to base a

log_a b = x means exactly a^x = b. Switch between the two forms, and see why log_a a = 1 and log_a 1 = 0.

Preview10m

The laws of logarithms

Multiplying numbers adds their logarithms, dividing subtracts them, and a power becomes a multiplier. Each law comes from an index law.

12m

Using the laws of logarithms

Write several logarithms as one, or split one into parts. A plain number such as 3 becomes lg 1000 first.

12m

Solving logarithmic equations

Combine the logarithms into one, rewrite it as a power, then solve. Reject any answer that puts a negative number inside a log.

12m

Solving exponential equations

Solve a^x = b by taking logarithms of both sides, so x = lg b / lg a. Give the answer to 3 significant figures.

12m

Solving exponential equations using a substitution

An equation such as 4^x - 6(2^x) + 8 = 0 becomes a quadratic when y = 2^x. A negative value of y gives no solution.

12m

Change of base of logarithms

log_a b = log_c b / log_c a for any base c. Use it to work out awkward bases and to solve equations with two different bases.

12m

Natural logarithms

The number e is about 2.718. The logarithm to base e is written ln, so ln x = y means e^y = x.

10m

10.

Practical applications of exponential equations

Growth and decay models such as N = 200e^(0.1t). Find the starting value, a later value, or the time taken to reach a target.

12m

11.

The graphs of exponential functions

Sketch y = ke^(nx) + a. The line y = a is an asymptote, and the signs of k and n decide the side and the direction.

12m

12.

The graphs of logarithmic functions

Sketch y = k ln(ax + b). It exists only where ax + b > 0, and it has a vertical asymptote where ax + b = 0.

12m

13.

The inverse of logarithmic and exponential functions

e^x and ln x undo each other, so their graphs are reflections in y = x. Find inverses such as that of f(x) = 3e^(2x) - 1.

12m

14.

Logarithmic and exponential functions: practice

A timed drill of mixed questions from the whole chapter, from reading a log to solving and sketching.

15m

15.

Logarithmic and exponential functions: exam practice

Paper 1 and Paper 2 style questions with full worked solutions and mark schemes.

15m

16.

Logarithmic and exponential functions: summary

The key facts, laws, graphs and common mistakes of the chapter on one screen.

10m

7.1-7.4 · Straight-line graphs

Find lengths, midpoints and equations of lines, use parallel and perpendicular gradients, and turn curved relationships into straight lines to find their unknown constants.

0/13

Length of a line

The length from (x1, y1) to (x2, y2) comes from Pythagoras: square the two differences, add them, then take the square root.

Preview10m 2.

Midpoint of a line

The midpoint is the average of the x-coordinates and the average of the y-coordinates. Work backwards to find a missing end.

Preview10m

Parallel and perpendicular lines

Parallel lines have equal gradients. Perpendicular gradients multiply to -1, so a gradient m becomes -1/m.

12m

Equations of straight lines

Use y - y1 = m(x - x1) with a gradient and a point, then tidy the result into y = mx + c or ax + by = c.

12m

The perpendicular bisector

Every point on the perpendicular bisector of AB is the same distance from A and B. Find the midpoint, flip the gradient, write the line.

12m

Solving problems with straight lines

Find where two lines meet, the foot of a perpendicular and a missing corner of a shape, using gradients, midpoints and lengths.

12m

Converting a non-linear equation to linear form

Choose new variables X and Y so that a curve such as y^2 = Ax^3 + B becomes the straight line Y = AX + B.

12m

Using logarithms to convert to linear form

Take logs of y = Ax^n or y = Ab^x. The result is a straight line whose gradient and intercept hold the constants.

12m

Converting from linear form to a non-linear equation

Two points on a transformed straight line give its gradient and intercept. Put them back to write y in terms of x.

12m

10.

Finding relationships from data

Turn a table of values into lg x and lg y, draw the line of best fit, then read off the constants A and n.

12m

11.

Straight-line graphs: practice

A timed drill of mixed questions from the whole chapter, from a midpoint to a transformed graph.

15m

12.

Straight-line graphs: exam practice

Paper 1 and Paper 2 style questions with full worked solutions and mark schemes.

15m

13.

Straight-line graphs: summary

The formulas, methods and common mistakes of the chapter on one screen.

10m

8.1-8.4 · Coordinate geometry of the circle

Write and read circle equations in both forms, find where a line meets a circle, find tangents, and decide how two circles meet and where their common chord lies.

0/12

The equation of a circle

Every point on a circle is r from the centre (a, b). Pythagoras turns that fact into (x - a)^2 + (y - b)^2 = r^2.

Preview10m 2.

The general form of the equation of a circle

Complete the square in x and in y to change x^2 + y^2 + 2gx + 2fy + c = 0 back into centre and radius form.

Preview12m

Finding the equation of a circle

Find the centre and the radius first: from a centre and a point, from the two ends of a diameter, or from other facts.

12m

Intersection of a line and a circle

Substitute the line into the circle equation and solve the quadratic. Each root gives one point where they meet.

12m

Tangent, chord or no intersection

The discriminant of that quadratic decides: two roots give a chord, one repeated root a tangent, no real roots a miss.

12m

Equations of tangents

A tangent is perpendicular to the radius at the point where it touches. Use the negative reciprocal of the radius gradient.

12m

Solving problems involving tangents

Find where two tangents meet, the length of a tangent from a point outside, and the value of k that makes a line a tangent.

12m

Intersection of two circles

Compare the distance between the centres with the sum and the difference of the radii: the circles cross, touch or do not meet.

12m

The common chord of two circles

Subtract one circle equation from the other. The squared terms cancel and leave the line of the common chord; solve with it.

12m

10.

Coordinate geometry of the circle: practice

A timed drill of mixed questions from the whole chapter, from reading a centre to finding a common chord.

15m

11.

Coordinate geometry of the circle: exam practice

Paper 1 and Paper 2 style questions with full worked solutions and mark schemes.

15m

12.

Coordinate geometry of the circle: summary

The circle forms, tests and methods of the chapter, with the common mistakes, on one screen.

10m

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.