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Comprehensive Guide to Logarithmic and Complex Number Equations

This guide covers essential mathematical concepts including logarithmic equations, complex numbers, and their properties. It explains the rules for manipulating logarithms and provides step-by-step solutions to complex equations, including exponential equations and Argand diagrams. Key topics include finding moduli of complex numbers, rationalizing denominators, and utilizing the factor theorem. This resource is ideal for students seeking to strengthen their understanding of mathematics through clear explanations and practical examples.

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Comprehensive Guide to Logarithmic and Complex Number Equations

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  1. COMPLEX JEAPORDY 10 10 10 10 20 20 20 20 30 30 30 30 100 100 100 50

  2. 10 points. Log Rules • Write as the log of a single number: • 3log(5)+2log(3)-log(5) log225

  3. 20 points. Exponential Equations • Solve 23x-1=12 X=1.528

  4. 30 points. Log equation • Solve ln(x+3)=ln(x)+ln(3) X=3/2 or 1.5

  5. 50 points. Solving with logs • Solve this for x in terms of k

  6. 10 points. Argand Diagrams, manipulation • What are the complex numbers a and c ? • Find ac where a and c are shown below: A=(1+2i) C=(-3+4i) AC=(-11-2i)

  7. 20 points. Modulus • Find the modulus of z where z=2+3i [z]=√13

  8. 30 points. Imaginary factors is a root of Find the value of A A=6

  9. 10 points. Simplify

  10. 20 points. Rationalise the denominator

  11. 30 points. Irrational Equation • Solve: X=8 (x=1 does not hold)

  12. 100 points. Harder solving • Solve the equation X=19.6 or 1.38

  13. 10 points. Use the factor theorem to factorise & hence solve (x-1)(x+2)(x+5)=0 X=1,-2 or -5

  14. 20 points. Solve the following equation by completing the square

  15. 30 points. Solving • One root of the equation is 2i. • Find the value of k and the two remaining roots. • f(2i)=0 since 2i is a root. By substitution (factor theorem) k must equal 32. • Roots are 2i, -2i (conjugate root theorem) and -8 (by evaluating constant)

  16. 100 points. Binomial Expansion Find the value of the term that is independent of x (the constant) in the expansion of: For r=8, the (9th) term is:

  17. 100 points. Algebraic Proof • Show that if w and z are two complex numbers. • Let w=x+yi and z=a+ib • LHS: (x-yi)(a+bi)=ax+bxi-ayi-byi2=ax+by+bxi-ayi=ax+by+(bx-ay)i • RHS: =LHS Therefore statement is true

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