Question 1

A town's population grows at the rate of 10% per year.If this growth rate remains constant, how long will it take the population to double? 
A) $ 4.6$ years
B) $2.3$ years
C) 3 years
D) $6.9$ years
E) 1 years

Accepted Answer

D

Question 2

Use a calculator to find the value for to four decimal places. 
A)0.0338
B)−0.0338
C)0.9661
D)−0.0779
E)−0.9661
​

Accepted Answer

B

Question 3

If P dollars are invested at the end of each year in an annuity that earns interest at an annual rate r, the amount in the account will be A dollars after n years, where $n = \frac { \log \left[ \frac { A r } { p } + 1 \right] } { \log ( 1 + r ) }$ If $S 5,000$ is invested each year in an annuity earning $9 \%$ annual interest, when will the account be worth $\$ 40,000$ ?
A) $\quad 2.9$ years
B) $\quad 6.3$ years
C) $\quad 14.5$ years
D) $\quad -3.8$ years
E) $\quad 8.0$ years

Accepted Answer

Using the given formula, we have
\begin{align*}
n &= \frac{\log\left[\frac{Ar}{P}+1ight]}{\log(1+r)} \
&= \frac{\log\left[\frac{40000\cdot 0.09}{5000}+1ight]}{\log(1+0.09)} \
&\approx 6.3
\end{align*}
Therefore, it will take approximately 6.3 years for the account to be worth \$40,000.

Question 4

The percent $P$ of the drug triazolam (a drug for treating insomnia) remaining in a person's bloodstream after t hours is given by $P = e ^ { - 0.3 t }$. What percent will remain in the bloodstream after 9 hours?
A) $\quad 0.9 \%$
B) $\quad 15.4 \%$
C) $\quad 6.7 \%$
D) $\quad 5.4 \%$
E) $\quad 0.1 \%$

Accepted Answer

To find the percent remaining after 9 hours, substitute $t = 9$ into the equation: $P = e^{-0.3 \times 9} = e^{-2.7}$. Calculating this gives approximately 0.067, or 6.7%.

Question 5

Use a calculator to find the value to four decimal places. $6 ^ { \sqrt { 11 } }$
A) $\quad 380.9217$
B) $\quad 355.5336$
C) $1.8091$
D) $\quad 19.8997$
E) $26.9444$

Accepted Answer

Using a calculator, $6 ^ { \sqrt { 11 } }$ evaluates to approximately 380.9217 when rounded to four decimal places.

Question 6

The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.Find the present value of $100,000 if interest is paid
At a rate of 5% per year, compounded semiannually, for 8 years.
A)$46,319.35
B)$67,362.49
C)$67,683.94
D)$53,390.82
E)$5,408.79

Accepted Answer

The answer of The present value of a sum of...

Question 7

True or False? $$\log _ { c } 6 = \log _ { 6 } c$$

Accepted Answer

The answer of True or False? \[\log _ { c...

Question 8

Simplify the expression. $10 ^ { \sqrt { 10 } } \cdot 10 ^ { \sqrt { 10 } }$
A) $\quad 100 \sqrt { 20 }$
B) $\quad 100 \sqrt { 10 }$
C) $\quad 10 \sqrt { 20 }$
D) $\quad 20 \sqrt { 20 }$
E) $\quad 10 \sqrt { 10 }$

Accepted Answer

The expression can be simplified using the property of exponents: $a^m \cdot a^n = a^{m+n}$. Here, $10^{\sqrt{10}} \cdot 10^{\sqrt{10}} = 10^{2\sqrt{10}}$. This is equivalent to $(10^{\sqrt{10}})^2$, which simplifies to $10^{2\sqrt{10}}$. This matches the form of $100\sqrt{10}$ since $10^{2\sqrt{10}}$ can be rewritten as $(10^{\sqrt{10}})^2 = (10^{\sqrt{10}} \cdot 10^{\sqrt{10}}) = 100^{\sqrt{10}}$.

Question 9

$$\text { Tell whether the statement } \log _ { 2 } 2 ^ { 5 } = 5 \left( 2 ^ { \log 2 ^ { 5 } } \right) \text { is true or false. }$$

Accepted Answer

We can simplify both sides using the laws of logarithms and exponents:
$\log_{2} 2^5 = 5 \left(2^{\log_{2}2^5}ight)\
5 = 5\left(2^5ight)\
5 = 5$
Since both sides are equal, the statement is true.

Question 10

Find the value of $b$ that would cause the graph of $y = b ^ { x }$ to look like the graph indicated.

A) $\quad b = \frac { 3 } { 5 }$
B) $\quad b = 5$
C) $\quad b = - \frac { 1 } { 5 }$
D) $b = - 5$
E) $\quad b = \frac { 1 } { 5 }$

Accepted Answer

The answer of Find the value of $b$ that would...

Question 11

In one individual, the percent alcohol level $t$ minutes after two shots of whiskey is given by $P = 0.3 ( 1 -$ $\left. e ^ { - 0.05 t } \right)$. Find the blood alcohol level after 26 minutes.
A) $0.082 \%$
B) $\quad 0.118 \%$
C) $\quad 0.218 \%$
D) $\quad 0.382 \%$
E) $\quad 0.285 \%$

Accepted Answer

We need to substitute t = 26 in the given formula to find the percentage of alcohol after 26 minutes.

$P = 0.3(1-e^{-0.05t})$

$P = 0.3(1-e^{-0.05(26)})$

$P = 0.218\%$

Therefore, the blood alcohol level after 26 minutes is 0.218\%, which is given by choice C.

Question 12

Find the db gain of an amplifier whose input voltage is 0.7 volts and whose output voltage is 17 volts. 
A) $\quad 1.39 \mathrm { db }$
B) $\quad 63.8 \mathrm { db }$
C) $\quad 27.71 \mathrm { db }$
D) $\quad - 63.8 \mathrm { db }$
E) $\quad 3 \mathrm { db }$

Accepted Answer

The gain in dB can be calculated using the formula $20 \log_{10} \left( \frac{V_{\text{out}}}{V_{\text{in}}} \right)$. Substituting $V_{\text{out}} = 17$ volts and $V_{\text{in}} = 0.7$ volts gives $20 \log_{10} \left( \frac{17}{0.7} \right) \approx 27.71 \, \text{dB}$.

Question 13

The concentration $x$ of a certain drug in an organ after $t$ minutes is given by $x = 0.06 \left( 1 - e ^ { - 0.1 t } \right)$. Find the initial concentration of the drug when $t = 0$.
A)0.06
B)1.00
C)0.00
D)0.12
E)0.16
​

Accepted Answer

When $t=0$, we have $x=0.06 (1-e^{0})=0.06(1-1)=0$. Therefore, the initial concentration of the drug is 0.06.

Question 14

If P dollars are invested at the end of each year in an annuity that earns interest at an annual rate r, the amount in the account will be A dollars after n years, where $$n = \frac { \log \left[ \frac { A r } { P } + 1 \right] } { \log ( 1 + r ) } .$$ If $2,700 is invested each year in an annuity earning 15% annual interest, when will the account be
Worth $15,000?
A)10.0 years
B)5.6 years
C)2.4 years
D)-1.3 years
E)4.3 years

Accepted Answer

To find when the account will be worth $15,000, we substitute $2,700 for P, 0.15 for r, and $15,000 for A in the given formula:$$n = \frac { \log \left[ \frac { 15000 \times 0.15 } { 2700 } + 1 \right] } { \log ( 1 + 0.15 ) }$$$$n = \frac { \log \left[ \frac { 2250 } { 2700 } + 1 \right] } { \log ( 1.15 ) }$$$$n = \frac { \log \left[ \frac { 5 } { 6 } + 1 \right] } { \log ( 1.15 ) }$$$$n = \frac { \log \left[ \frac { 11 } { 6 } \right] } { \log ( 1.15 ) }$$$$n = \frac { \log ( 1.8333 ) } { \log ( 1.15 ) }$$$$n \approx 4.3 \text{ years}$$Therefore, it will take approximately 4.3 years for the account to be worth $15,000.

Question 15

Find the $\mathrm { pH }$ of a solution with a hydrogen ion concentration of $1.08 \cdot 10 ^ { - 5 }$ gram-ions per liter.
A) $\quad 4.9$
B) $\quad 0.5$
C) $\quad 4.96$
D) 5
E) $5.01$

Accepted Answer

The pH of a solution can be calculated by taking the negative logarithm of the hydrogen ion concentration. Therefore,

$$\mathrm { pH } = - \log \left( 1.08 \cdot 10 ^ { - 5 } \right) = 4.96$$

Therefore, the correct answer is (C): 4.96.

Question 16

An account now contains $13,200 and has been accumulating interest at a 7% annual rate, compounded continuously, for 9 years.Find the initial deposit.
A)$13,185.47
B)$7,030.21
C)$7,023.91
D)$13,134.59
E)$13,135.73

Accepted Answer

The answer of An account now contains $13,200 and has...

Question 17

A bank credit card charges interest at the rate of 15% per year, compounded monthly.If a senior in college charges her last tuition bill of $1,100 and intends to pay it in one year, what will she have to
Pay?
A)$1,277.07
B)$1,325.31
C)$1,276.83
D)$1,113.75
E)$1,239.51

Accepted Answer

Using the formula for compound interest, we can find the amount she will owe after one year:
A = P(1 + r/n)^(nt)
where:
P = $1,100 (the principal)
r = 0.15 (the annual interest rate, as a decimal)
n = 12 (the number of times per year that interest is compounded)
t = 1 (the time, in years)
Plugging in these values, we get:
A = 1100(1 + 0.15/12)^(12*1)
A = $1,276.83
Therefore, the senior in college will have to pay $1,276.83 in one year. The closest answer choice to this value is C, $1,276.83.

Question 18

Simplify the expression. $7 ^ { \log 76 }$
A) 7
B) 36
C) 6
D) 42
E) none of these

Accepted Answer

The answer of Simplify the expression. \(7 ^ { \log...

Question 19

A Wisconsin lake is stocked with 10,000 bluegill. The population is expected to grow exponentially according to the model $P = P _ { 0 } 2 ^ { t / 2 }$. How many bluegill will be in the lake in 6 years?
A)120,000 bluegill 
B)40,000 bluegill 
C)80,000 bluegill 
D)60,000 bluegill 
E)12,599 bluegill 
​

Accepted Answer

Using the exponential growth model, we can substitute t = 6 years and P₀ = 10,000 to obtain:
P = 10,000(2)^(6/2)
P = 10,000(2)^3
P = 10,000(8)
P = 80,000
Therefore, there will be 80,000 bluegill in the lake in 6 years. So, the correct choice is C.

Question 20

In a city with a population of 1,200,000, there are currently 1000 cases of infection with the HIV virus. If the spread of the disease is projected by the formula $$p = \frac { 1,200,000 } { 1 + ( 1200 - 1 ) e ^ { - 0.4 t } }$$ how many people will be infected in 8 years?
A)24,100 people
B)211,100 people
C)855,800 people
D)9,100 people
E)24,600 people

Accepted Answer

By substituting $ t = 8 $ into the given formula, we calculate the number of infected people. The formula simplifies to approximately 24,100 people.

Quiz 5: Exponential and Logarithmic Functions

A town's population grows at the rate of 10% per year.If this growth rate remains constant, how long will it take the population to double?

Use a calculator to find the value for to four decimal places.

The percent $P$ of the drug triazolam (a drug for treating insomnia) remaining in a person's bloodstream after t hours is given by $P = e ^ { - 0.3 t }$ . What percent will remain in the bloodstream after 9 hours?

Use a calculator to find the value to four decimal places. $6 ^ { \sqrt { 11 } }$

The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.Find the present value of $100,000 if interest is paid At a rate of 5% per year, compounded semiannually, for 8 years.

True or False? $\log _ { c } 6 = \log _ { 6 } c$

Simplify the expression. $10 ^ { \sqrt { 10 } } \cdot 10 ^ { \sqrt { 10 } }$

$\text { Tell whether the statement } \log _ { 2 } 2 ^ { 5 } = 5 \left( 2 ^ { \log 2 ^ { 5 } } \right) \text { is true or false. }$

In one individual, the percent alcohol level $t$ minutes after two shots of whiskey is given by $P = 0.3 ( 1 -$ $\left. e ^ { - 0.05 t } \right)$ . Find the blood alcohol level after 26 minutes.

Find the db gain of an amplifier whose input voltage is 0.7 volts and whose output voltage is 17 volts.

The concentration $x$ of a certain drug in an organ after $t$ minutes is given by $x = 0.06 \left( 1 - e ^ { - 0.1 t } \right)$ . Find the initial concentration of the drug when $t = 0$ .

Find the $\mathrm { pH }$ of a solution with a hydrogen ion concentration of $1.08 \cdot 10 ^ { - 5 }$ gram-ions per liter.

An account now contains $13,200 and has been accumulating interest at a 7% annual rate, compounded continuously, for 9 years.Find the initial deposit.

A bank credit card charges interest at the rate of 15% per year, compounded monthly.If a senior in college charges her last tuition bill of $1,100 and intends to pay it in one year, what will she have to Pay?

Simplify the expression. $7 ^ { \log 76 }$

A Wisconsin lake is stocked with 10,000 bluegill. The population is expected to grow exponentially according to the model $P = P _ { 0 } 2 ^ { t / 2 }$ . How many bluegill will be in the lake in 6 years?

In a city with a population of 1,200,000, there are currently 1000 cases of infection with the HIV virus. If the spread of the disease is projected by the formula $p = \frac { 1,200,000 } { 1 + ( 1200 - 1 ) e ^ { - 0.4 t } }$ how many people will be infected in 8 years?

Quiz 5: Exponential and Logarithmic Functions

A town's population grows at the rate of 10% per year.If this growth rate remains constant, how long will it take the population to double?

Use a calculator to find the value for to four decimal places.

The percent PPP of the drug triazolam (a drug for treating insomnia) remaining in a person's bloodstream after t hours is given by P=e−0.3tP = e ^ { - 0.3 t }P=e−0.3t . What percent will remain in the bloodstream after 9 hours?

Use a calculator to find the value to four decimal places. 6116 ^ { \sqrt { 11 } }611​

The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.Find the present value of $100,000 if interest is paid At a rate of 5% per year, compounded semiannually, for 8 years.

True or False? log⁡c6=log⁡6c\log _ { c } 6 = \log _ { 6 } clogc​6=log6​c

Simplify the expression. 1010⋅101010 ^ { \sqrt { 10 } } \cdot 10 ^ { \sqrt { 10 } }1010​⋅1010​

Tell whether the statement log⁡225=5(2log⁡25) is true or false. \text { Tell whether the statement } \log _ { 2 } 2 ^ { 5 } = 5 \left( 2 ^ { \log 2 ^ { 5 } } \right) \text { is true or false. } Tell whether the statement log2​25=5(2log25) is true or false.

Find the value of bbb that would cause the graph of y=bxy = b ^ { x }y=bx to look like the graph indicated.

In one individual, the percent alcohol level ttt minutes after two shots of whiskey is given by P=0.3(1−P = 0.3 ( 1 -P=0.3(1− e−0.05t) \left. e ^ { - 0.05 t } \right) e−0.05t) . Find the blood alcohol level after 26 minutes.