Question 1 How long does it take for an investment to double in value if it is invested at 14% compounded quarterly and compounded continuously? a) At 14% compounded quarterly, the investment doubles in how many years? b) At 14% compounded continuously, the investment doubles in how many years? Answer 1 Step 1 For compound interest, the formula is: 𝐴 = 𝑃 ( 1 + 𝑟 𝑛 ) 𝑛 𝑡 A=P(1+ n r ) nt where: 𝐴 A is the future value, 𝑃 P is the principal amount, 𝑟 r is the annual interest rate, 𝑛 n is the number of times interest is compounded per year, 𝑡 t is the time in years. Part (a): Compounded Quarterly Let 𝐴 = 2 𝑃 A=2P (since we want to double the investment), 𝑟 = 0.14 r=0.14, and 𝑛 = 4 n=4 (quarterly compounding). 2 𝑃 = 𝑃 ( 1 + 0.14 4 ) 4 𝑡 2P=P(1+ 4 0.14 ) 4t Dividing by 𝑃 P: 2 = ( 1.035 ) 4 𝑡 2=(1.035) 4t Taking the natural logarithm on both sides: ln ⁡ ( 2 ) = 4 𝑡 ⋅ ln ⁡ ( 1.035 ) ln(2)=4t⋅ln(1.035) Solving for 𝑡 t: 𝑡 = ln ⁡ ( 2 ) 4 ⋅ ln ⁡ ( 1.035 ) ≈ 5.04 years t= 4⋅ln(1.035) ln(2) ≈5.04 years Part (b): Compounded Continuously For continuous compounding, the formula is: 𝐴 = 𝑃 𝑒 𝑟 𝑡 A=Pe rt Using 𝐴 = 2 𝑃 A=2P, 𝑟 = 0.14 r=0.14: 2 𝑃 = 𝑃 𝑒 0.14 𝑡 2P=Pe 0.14t Dividing by 𝑃 P: 2 = 𝑒 0.14 𝑡 2=e 0.14t Taking the natural logarithm on both sides: ln ⁡ ( 2 ) = 0.14 𝑡 ln(2)=0.14t Solving for 𝑡 t: 𝑡 = ln ⁡ ( 2 ) 0.14 ≈ 4.95 years t= 0.14 ln(2) ≈4.95 years So, the answers are: a) 5.04 years for quarterly compounding, b) 4.95 years for continuous compounding. Question 2 Solve the equation log ⁡ 8 ( 𝑥 + 5 ) − log ⁡ 8 ( 𝑥 − 2 ) = 1 log 8 (x+5)−log 8 (x−2)=1. Answer 1 Using the properties of logarithms: log ⁡ 8 ( 𝑥 + 5 𝑥 − 2 ) = 1 log 8 ( x−2 x+5 )=1 Rewrite in exponential form: 𝑥 + 5 𝑥 − 2 = 8 x−2 x+5 =8 Multiply both sides by 𝑥 − 2 x−2: 𝑥 + 5 = 8 ( 𝑥 − 2 ) x+5=8(x−2) Expanding and simplifying: 𝑥 + 5 = 8 𝑥 − 16 x+5=8x−16 Rearrange to isolate 𝑥 x: 5 + 16 = 8 𝑥 − 𝑥 5+16=8x−x 21 = 7 𝑥 21=7x 𝑥 = 3 x=3 Thus, the exact solution is 𝑥 = 3 x=3. Question 3 Solve the equation log ⁡ 3 ( 𝑥 − 8 ) + log ⁡ 3 𝑥 = 2 log 3 (x−8)+log 3 x=2. Answer 1 Combine the logarithms: log ⁡ 3 ( ( 𝑥 − 8 ) 𝑥 ) = 2 log 3 ((x−8)x)=2 Rewrite in exponential form: ( 𝑥 − 8 ) 𝑥 = 3 2 (x−8)x=3 2 𝑥 2 − 8 𝑥 = 9 x 2 −8x=9 Rearrange to form a quadratic equation: 𝑥 2 − 8 𝑥 − 9 = 0 x 2 −8x−9=0 Factor the quadratic: ( 𝑥 − 9 ) ( 𝑥 + 1 ) = 0 (x−9)(x+1)=0 So, 𝑥 = 9 x=9 or 𝑥 = − 1 x=−1. Since 𝑥 x must be positive for the logarithm to be defined, we discard 𝑥 = − 1 x=−1. Thus, the exact solution is 𝑥 = 9 x=9.