Question

question 1 solve 144 = 9(4^x)
question 2 solve 4 = log8(y + 5)
question 3 solve 20747 = 12^x + 11
question 4 solve 81^(x - 8)=6561^(2x - 1)
question 5 cell phone usage in canada has been increasing exponentially at 20% per year. how many people originally used cell phones if after 5 years there were 373248 users?
question 6 the number of octopi in the ocean has been exponentially decreasing at a rate of 80% per year. how long did it take the population to decrease from 2000000 octopi to 640 octopi?
question 7 the half - life of uranium is 42 years. after 462 years, there is still 600 kg remaining in the sea. how much was there originally?
question 8 a population doubles every 7 years. how long will it take the population to grow from 200 people to 12800?

Explanation:

Step1: Solve Question 1

First, divide both sides of $144 = 9(4^{x})$ by 9. We get $\frac{144}{9}=4^{x}$, so $16 = 4^{x}$. Since $16 = 4^{2}$, then $x = 2$.

Step2: Solve Question 2

Rewrite the logarithmic equation $4=\log_{8}(y + 5)$ in exponential form. We have $8^{4}=y + 5$. Calculate $8^{4}=4096$, then $y=4096 - 5=4091$.

Step3: Solve Question 3

Subtract 11 from both sides of $20747=12^{x}+11$. We get $20747-11 = 12^{x}$, so $20736=12^{x}$. Since $12^{4}=20736$, then $x = 4$.

Step4: Solve Question 4

Rewrite $81^{x - 8}=6561^{2x - 1}$. Since $81 = 3^{4}$ and $6561=3^{8}$, we have $(3^{4})^{x - 8}=(3^{8})^{2x - 1}$. Using the power - of - a - power rule, $3^{4(x - 8)}=3^{8(2x - 1)}$. Then $4(x - 8)=8(2x - 1)$. Expand: $4x-32 = 16x-8$. Rearrange terms: $16x-4x=-32 + 8$, $12x=-24$, so $x=-2$.

Step5: Solve Question 5

Use the exponential growth formula $A = P(1 + r)^{t}$, where $A = 373248$, $r=0.2$, and $t = 5$. We have $373248=P(1 + 0.2)^{5}$. First, calculate $(1 + 0.2)^{5}=1.2^{5}=2.48832$. Then $P=\frac{373248}{2.48832}=150000$.

Step6: Solve Question 6

Use the exponential decay formula $A = P(1 - r)^{t}$, where $P = 2000000$, $A = 640$, and $r = 0.8$. So $640=2000000(1 - 0.8)^{t}$. First, simplify to $\frac{640}{2000000}=(0.2)^{t}$, $0.00032=(0.2)^{t}$. Since $0.2=\frac{1}{5}$ and $0.00032=\frac{32}{100000}=\frac{1}{3125}=(\frac{1}{5})^{5}$, then $t = 5$.

Step7: Solve Question 7

Use the half - life formula $A=P(\frac{1}{2})^{\frac{t}{h}}$, where $A = 600$, $h = 42$, and $t = 462$. First, calculate $\frac{t}{h}=\frac{462}{42}=11$. Then $600=P(\frac{1}{2})^{11}$. Solve for $P$: $P=600\times2^{11}=600\times2048 = 1228800$.

Step8: Solve Question 9

Use the exponential growth formula $A = P(2)^{\frac{t}{k}}$, where $P = 200$, $A = 12800$, and $k = 7$. We have $12800=200\times2^{\frac{t}{7}}$. First, divide both sides by 200: $\frac{12800}{200}=2^{\frac{t}{7}}$, $64 = 2^{\frac{t}{7}}$. Since $64 = 2^{6}$, then $\frac{t}{7}=6$, and $t = 42$.

Answer:

Question 1: $x = 2$
Question 2: $y = 4091$
Question 3: $x = 4$
Question 4: $x=-2$
Question 5: 150000
Question 6: 5 years
Question 7: 1228800 kg
Question 8: 42 years