Question 1

Solve the equation.
-$125 ^ { x - 2 } = 25$

Accepted Answer

First, express both sides of the equation in terms of a common base. Since $125 = 5^3$ and $25 = 5^2$, the equation becomes $5^{3(x-2)} = 5^2$. Equating the exponents gives $3(x - 2) = 2$, which simplifies to $x = \frac{8}{3}$.

Question 2

Solve the equation.
-$$\left( \frac { 1 } { 4 } \right) ^ { x } = 8 ^ { 2 - x }$$

Accepted Answer

The answer of Solve the equation.
-\[\left( \frac { 1 }...

Question 3

Solve. Round the answer to the nearest whole.
-The size of the rat population of a wharf area grows at a rate of 7% monthly. If there are 300 rats in June, find how many rats should be expected by next June.

Accepted Answer

The answer of Solve. Round the answer to the nearest...

Question 4

Solve. Round the answer to the nearest whole.
-Suppose a city with population 320,000 has been growing at a rate of 4% per year. If this rate continues, find the population of this city in 15 years.

Accepted Answer

The population can be calculated using the formula for compound growth: P = P0 * (1 + r)^t, where P0 is the initial population, r is the growth rate, and t is the time in years. Here, P0 = 320,000, r = 0.04, and t = 15. Plugging in these values, P = 320,000 * (1 + 0.04)^15 ≈ 576,302.

Question 5

Solve.
-The amount of a radioactive substance present, in grams, at time t in months is given by the formula y = 7000(3)-0.2t. Find the number of grams present in 3 years. If necessary, round to three decimal places.

Accepted Answer

The answer of Solve.
-The amount of a radioactive substance present,...

Question 6

Solve.
-The rabbit population in a forest area grows at the rate of 6% monthly. If there are 300 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. $$\text { Use } y = 300 ( 2.7 ) ^ { 0.06 t }$$

Accepted Answer

The formula given is incorrect for this problem. The correct formula for exponential growth is $ y = 300 \times (1 + 0.06)^{12} $. Calculating this gives approximately 613 rabbits.

Question 7

Use the exponential decay formula to find the final amount. Round to the nearest whole.
-$\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Original } \
\text { Amount }
\end{array} & \begin{array}{c}
\text { Decay } \
\text { Rate per Year }
\end{array} & \begin{array}{c}
\text { Number } \
\text { of Years, } \times
\end{array} & \begin{array}{c}
\text { Final Amount after } \
\text { x Years of Decay }
\end{array} \
\hline 207 & 4 \% & 7 &
\end{array}$

Accepted Answer

The exponential decay formula is $A = P(1 - r)^t$, where $A$ is the final amount, $P$ is the original amount, $r$ is the decay rate per period, and $t$ is the number of periods. Plugging in the given values: $A = 207(1 - 0.04)^7 \approx 156$.

Question 8

Use the exponential decay formula to find the final amount. Round to the nearest whole.
-

Accepted Answer

The answer of Use the exponential decay formula to find...

Question 9

Use the exponential growth formula to find the final amount. Round to the nearest whole.
-$\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Original } \
\text { Amount }
\end{array} & \begin{array}{c}
\text { Growth } \
\text { Rate per Year }
\end{array} & \begin{array}{c}
\text { Number } \
\text { of Years, }
\end{array} & \begin{array}{l}
\text { Final Amount after } \
\text { x Years of Growth }
\end{array} \
\hline 26 & 64 \% & 16 &
\end{array}$

Accepted Answer

The answer of Use the exponential growth formula to find...

Question 10

Use the exponential growth formula to find the final amount. Round to the nearest whole.
-

Accepted Answer

The answer of Use the exponential growth formula to find...

Question 11

Use the exponential decay formula with half-lives to find the final amount. Round to the nearest tenth when necessary.
-$\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Original } \
\text { Amount }
\end{array} & \begin{array}{c}
\text { Half-Life } \
\text { (in years) }
\end{array} & \begin{array}{c}
\text { Number } \
\text { of Years }
\end{array} & \begin{array}{c}
\text { Final Amount after } \
\text { x Time Intervals }
\end{array} \
\hline 26 & 125 & 800 &
\end{array}$

Accepted Answer

The exponential decay formula is $A = P(1/2)^{t/T}$, where $A$ is the final amount, $P$ is the original amount, $t$ is the time elapsed, and $T$ is the half-life. Substituting $P = 26$, $t = 800$, and $T = 125$, we get $A = 26(1/2)^{800/125} = 26(1/2)^{6.4} \approx 0.3$.

Question 12

Use the exponential growth formula to find the final amount. Round to the nearest whole.
-

Accepted Answer

The answer of Use the exponential growth formula to find...

Question 13

Solve.
-A city is growing at the rate of 0.8% annually. If there were 3,917,000 residents in the city in 1994, find how many (to the nearest ten-thousand) are living in that city in 2000 $$\text { Use } y = 3,917,000 ( 2.7 ) ^ { 0.008 t } \text {. }$$

Accepted Answer

The answer of Solve.
-A city is growing at the rate...

Question 14

Use the exponential decay formula with half-lives to find the final amount. Round to the nearest tenth when necessary.
-$\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Original } \
\text { Amount }
\end{array} & \begin{array}{c}
\text { Half-Life } \
\text { (in years) }
\end{array} & \begin{array}{c}
\text { Number } \
\text { of Years }
\end{array} & \begin{array}{c}
\text { Final Amount after } \
\text { x Time Intervals }
\end{array} \
\hline 500 & 15 & 45 &
\end{array}$

Accepted Answer

The exponential decay formula is $A = P(1/2)^{t/T}$, where $A$ is the final amount, $P$ is the original amount, $t$ is the number of years, and $T$ is the half-life in years. Plugging in the given values: $A = 500(1/2)^{45/15} = 500(1/2)^3 = 500(1/8) = 62.5$.

Question 15

Use the exponential decay formula to find the final amount. Round to the nearest whole.
-$\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Original } \
\text { Amount }
\end{array} & \begin{array}{c}
\text { Decay } \
\text { Rate per Year }
\end{array} & \begin{array}{c}
\text { Number } \
\text { of Years, }
\end{array} & \begin{array}{c}
\text { Final Amount after } \
\text { x Years of Decay }
\end{array} \
\hline 12,000 & 12 \% & 11 &
\end{array}$

Accepted Answer

The exponential decay formula is $A = P(1 - r)^t$, where $A$ is the amount after decay, $P$ is the original amount, $r$ is the decay rate per period, and $t$ is the number of periods. Plugging in the given values: $A = 12000(1 - 0.12)^{11}$, calculating this gives approximately 2941.

Question 16

Solve.
-Four bacteria are placed in a petri dish. The population will double every day. The formula for the number of bacteria in the dish on day t is $$\mathrm { N } ( \mathrm { t } ) = 4 ( 2 ) ^ { \mathrm { t } }$$ where t is the number of days after the four bacteria are placed in the dish. How many bacteria are in the dish six
Days after the four bacteria are placed in the dish?

Accepted Answer

To find the number of bacteria in the dish six days after the initial placement, substitute $t = 6$ into the given formula: $\mathrm{N}(6) = 4(2)^6 = 4 \times 64 = 256$.

Question 17

Use the exponential growth formula to find the final amount. Round to the nearest whole.
-

Accepted Answer

The answer of Use the exponential growth formula to find...

Question 18

Solve.
-Hector borrows $3750 at a rate of 9% compounded semiannually. Find how much Hector owes at the end of 5 years. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$. Round to two decimal places.

Accepted Answer

Using the compound interest formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$, where $P = 3750$, $r = 0.09$, $n = 2$ (since it's compounded semiannually), and $t = 5$, we calculate $A = 3750 \left( 1 + \frac { 0.09 } { 2 } \right) ^ { 2 \times 5 }$ which equals approximately \$5823.64.

Question 19

Use the exponential decay formula with half-lives to find the final amount. Round to the nearest tenth when necessary.
-$\begin{array}{c|c|c|c}
\text { Original } & \text { Half-Life } \
\text { Amount } & \text { (in years) } & \begin{array}{c}
\text { Number } \
\text { of Years }
\end{array} & \begin{array}{c}
\text { Final Amount after } \
\text { x Time Intervals }
\end{array} \
\hline 400 & 8 & 52 &
\end{array}$

Accepted Answer

The exponential decay formula is $A = P(1/2)^{t/T}$, where $A$ is the final amount, $P$ is the original amount, $t$ is the time elapsed, and $T$ is the half-life period. Plugging in the given values: $A = 400(1/2)^{52/8} = 400(1/2)^{6.5} \approx 4.4$.

Question 20

Use the exponential decay formula with half-lives to find the final amount. Round to the nearest tenth when necessary.
-$\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Original } \
\text { Amount }
\end{array} & \begin{array}{c}
\text { Half-Life } \
\text { (in years) }
\end{array} & \begin{array}{c}
\text { Number } \
\text { of Years }
\end{array} & \begin{array}{c}
\text { Final Amount after } \
\text { x Time Intervals }
\end{array} \
\hline 60 & 6 & 12 &
\end{array}$

Accepted Answer

The exponential decay formula is $A = P(1/2)^{t/T}$, where $A$ is the amount remaining, $P$ is the initial amount, $t$ is the time elapsed, and $T$ is the half-life period. Plugging in the given values: $A = 60(1/2)^{12/6} = 60(1/2)^2 = 60(1/4) = 15$.

Quiz 9: Exponential and Logarithmic Functions

Solve the equation. - $125 ^ { x - 2 } = 25$

Solve the equation. - $\left( \frac { 1 } { 4 } \right) ^ { x } = 8 ^ { 2 - x }$

Solve. Round the answer to the nearest whole. -The size of the rat population of a wharf area grows at a rate of 7% monthly. If there are 300 rats in June, find how many rats should be expected by next June.

Solve. Round the answer to the nearest whole. -Suppose a city with population 320,000 has been growing at a rate of 4% per year. If this rate continues, find the population of this city in 15 years.

Solve. -The amount of a radioactive substance present, in grams, at time t in months is given by the formula y = 7000(3) -0.2t. Find the number of grams present in 3 years. If necessary, round to three decimal places.

Solve. -The rabbit population in a forest area grows at the rate of 6% monthly. If there are 300 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. $\text { Use } y = 300 ( 2.7 ) ^ { 0.06 t }$

Use the exponential decay formula to find the final amount. Round to the nearest whole. -

Use the exponential growth formula to find the final amount. Round to the nearest whole. -

Use the exponential growth formula to find the final amount. Round to the nearest whole. -

Solve. -A city is growing at the rate of 0.8% annually. If there were 3,917,000 residents in the city in 1994, find how many (to the nearest ten-thousand) are living in that city in 2000 $\text { Use } y = 3,917,000 ( 2.7 ) ^ { 0.008 t } \text {. }$

Use the exponential growth formula to find the final amount. Round to the nearest whole. -

Solve. -Hector borrows $3750 at a rate of 9% compounded semiannually. Find how much Hector owes at the end of 5 years. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ . Round to two decimal places.

Quiz 9: Exponential and Logarithmic Functions

Solve the equation. - 125x−2=25125 ^ { x - 2 } = 25125x−2=25

Solve the equation. - (14) x=82−x\left( \frac { 1 } { 4 } \right) ^ { x } = 8 ^ { 2 - x }(41​) x=82−x

Solve. Round the answer to the nearest whole. -The size of the rat population of a wharf area grows at a rate of 7% monthly. If there are 300 rats in June, find how many rats should be expected by next June.

Solve. Round the answer to the nearest whole. -Suppose a city with population 320,000 has been growing at a rate of 4% per year. If this rate continues, find the population of this city in 15 years.

Solve. -The amount of a radioactive substance present, in grams, at time t in months is given by the formula y = 7000(3) -0.2t. Find the number of grams present in 3 years. If necessary, round to three decimal places.

Solve. -The rabbit population in a forest area grows at the rate of 6% monthly. If there are 300 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. Use y=300(2.7) 0.06t\text { Use } y = 300 ( 2.7 ) ^ { 0.06 t } Use y=300(2.7) 0.06t

Use the exponential decay formula to find the final amount. Round to the nearest whole. -

Use the exponential growth formula to find the final amount. Round to the nearest whole. -

Use the exponential growth formula to find the final amount. Round to the nearest whole. -

Use the exponential growth formula to find the final amount. Round to the nearest whole. -

Solve. -Hector borrows $3750 at a rate of 9% compounded semiannually. Find how much Hector owes at the end of 5 years. Use: A=P(1+rn) ntA = P \left( 1 + \frac { r } { n } \right) ^ { n t }A=P(1+nr​) nt . Round to two decimal places.

Solve the equation. - $125 ^ { x - 2 } = 25$

Solve the equation. - $\left( \frac { 1 } { 4 } \right) ^ { x } = 8 ^ { 2 - x }$

Solve. -The rabbit population in a forest area grows at the rate of 6% monthly. If there are 300 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. $\text { Use } y = 300 ( 2.7 ) ^ { 0.06 t }$

Solve. -Hector borrows $3750 at a rate of 9% compounded semiannually. Find how much Hector owes at the end of 5 years. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ . Round to two decimal places.