Test Answers

Answers

Q1.   3^x+2  = 45 + 4 × 3^x  

Put all terms with 3^x  on the left, 3^x+2   – 4 × 3^x    = 45

Factor out  3^x  ,    3^x  (3²   – 4 )  = 45

…    Solve to get x = 2

Q2.     2 log₅r – 3 log ₁₂₅t = 2

Recall that 125 = 5³, so try to get like terms on the left.  This means use the change of base formula.

2 log₅r – 3 (log ₅t / log ₅125) = 2

2 log₅r – 3 (log ₅t / log ₅5³) = 2

Next remember that log _xx = 1, so log ₅5= 1

2 log₅r – 3 (log ₅t / log ₅5³) = 2

Next the 3s will cancel

2 log₅r – 3 (log ₅t / 3) = 2

Replace 2 with log ₅ 25 and you are almost finished

…  r = 5√t

Remember AB ≠ BA

So x = 3 and y = -5 .  (Confirm by substituting into both equations).

6.    4^x+2 × 2^y = 8

2, 4 and 8 are related by 4 = 2² and 8 = 2³.

4^x+2 × 2^y = 8  à(2²) ^x+2 × 2^y = 2³ à 2(x+2) + y = 3  Eqn (1)

Likewise, 27^{x – 2}  × (1/3)^y = 1 à (3³) ^{x – 2}  × (3) ^{– y} = 3⁰ à Eqn (2)

Don’t forget, x⁰ = 1   (unless 0⁰)

… Solve for the simultaneous equations: x = 1, y =  – 3.