Question 1

Solve the equation. $\log ( x - 46 ) - \log 6 = \log ( x - 32 ) - \log x$
A) $x = 52$
B) $\quad x = 4$
C) none of these
D) $x = 50$
E) $x = 48$

Accepted Answer

The answer of Solve the equation. \(\log ( x -...

Question 2

True or False? $$\log _ { a } 7 = \log _ { 7 } a$$

Accepted Answer

$\log_{a}7$ represents the exponent to which we must raise $a$ to get $7$, whereas $\log_{7}a$ represents the exponent to which we must raise $7$ to get $a$. These two exponents are generally not equal unless $a=7$. Therefore, the statement is false.

Question 3

A bank credit card charges interest at the rate of 16% per year, compounded monthly.If a senior in college charges her last tuition bill of $1,700 and intends to pay it in one year, what will she have to
Pay?
A)$2,101.29
B)$1,722.67
C)$1,915.60
D)$1,993.38
E)$1,992.86

Accepted Answer

To calculate the amount she will have to pay in one year, we use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount ($1,700), r is the annual interest rate (16% or 0.16), n is the number of times that interest is compounded per year (12, since it's compounded monthly), and t is the time the money is invested or borrowed for, in years (1 year). Plugging in the values, we get A = 1700(1 + 0.16/12)^(12*1) = $1,992.86.

Question 4

Solve the equation. $\log _ { 5 } x = \log _ { 5 } \frac { 1 } { x } + 8$
A) $\quad x = 637$
B) $\quad x = - 625$
C) $\quad x = - 637$
D) $\quad x = 626$
E) $\quad x = - 626$
F) $x = 625$

Accepted Answer

Using the property of logarithms, we can simplify the equation as follows:
\begin{align*}
\log_5 x &= \log_5 \frac{1}{x} + 8\
\log_5 x &= \log_5 1 - \log_5 x + 8\
2\log_5 x &= 9\
\log_5 x &= \frac{9}{2}
\end{align*}
Therefore, $x = 5^{9/2} = 625$.
Note that all the other choices for $x$ are either negative or have absolute value greater than $625$, which would result in a negative value for $\log_5 \frac{1}{x}$, violating the domain of logarithmic functions.

Question 5

In one individual, the percent alcohol level $t$ minutes after two shots of whiskey is given by $P = 0.3 ( 1 -$ $e ^ { - 0.05 t }$ ). Find the blood alcohol level after 29 minutes.
A) $\quad 0.370 \%$
B) $\quad 0.285 \%$
C) $\quad 0.230 \%$
D) $\quad 0.070 \%$
E) $\quad 0.118 \%$

Accepted Answer

We are given the function $P = 0.3 (1-e^{-0.05t})$ for the percent alcohol level after $t$ minutes. To find the blood alcohol level after 29 minutes, we simply substitute $t=29$ into the equation:
\begin{align*}
P &= 0.3 (1-e^{-0.05(29)}) \
&\approx 0.230\%
\end{align*}
Rounding to three decimal places, the blood alcohol level after 29 minutes is approximately 0.230\%, which corresponds to choice (C).

Question 6

$$\text { Tell whether the statement } \log _ { 7 } 7 ^ { 6 } = 6 \left( 7 ^ { \log 7 ^ { 6 } } \right) \text { is true or false. }$$

Accepted Answer

The statement is false because $\log_{7} 7^{6} = 6$, but $7^{\log 7^{6}} = 7^{6}$, so the equation simplifies to $6 \times 7^{6}$, which is not equal to 6.

Question 7

Simplify the expression. $5 ^ { \log 57 }$
A) 7
B) none of these
C) 49
D) 35
E) 5

Accepted Answer

The answer of Simplify the expression. \(5 ^ { \log...

Question 8

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 $\$ 7,000$ is invested each year in an annuity earning $6 \%$ annual interest, when will the account be worth $\$ 55,000 ?$
A) $\quad 15.3$ years
B) $\quad 6.6$ years
C) $\quad 3.2$ years
D) $\quad - 12.9$ years
E) $\quad 7.9$ years

Accepted Answer

The answer of If P dollars are invested at the...

Question 9

Solve the equation. $2 \log _ { 2 } x = 1 + \log _ { 2 } ( x + 40 )$
A) $x = 14$
B) $x = 10$
C) none of these
D) $x = 8$
E) $x = - 8$

Accepted Answer

The answer of Solve the equation. \(2 \log _ {...

Question 10

The concentration $x$ of a certain drug in an organ after $t$ minutes is given by $x = 0.08 \left( 1 - e ^ { - 0.1 t } \right)$. Find the initial concentration of the drug when $t = 0$.
A)0.16
B)0.00
C)0.08
D)1.00
E)0.22
​

Accepted Answer

When $t=0$, we have:
$$x=0.08\left(1-e^{-0.1(0)}\right)=0.08(1-e^0)=0.08\times0=0$$
Therefore, the initial concentration of the drug when $t=0$ is $0.08\times0 =0$, so the answer is (B). However, answer choice (B) is not an option in the given list, so the closest option would be (C), which is 0.08.

Question 11

An account now contains $11,300 and has been accumulating interest at a 9% annual rate, compounded continuously, for 9 years.Find the initial deposit.
A)$5,026.90
B)$11,209.29
C)$11,207.59
D)$11,279.45
E)$5,021.11

Accepted Answer

The initial deposit is the present value of the account, which can be calculated using the formula:PV = FV / e^(rt)where:PV is the present valueFV is the future valuer is the annual interest ratet is the number of yearsIn this case, the future value is $11,300, the annual interest rate is 9%, and the number of years is 9. Plugging these values into the formula, we get:PV = 11300 / e^(0.09 * 9)PV = 11300 / e^0.81PV = 11300 / 2.255PV = $5,026.90Therefore, the initial deposit was $5,026.90.

Question 12

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

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

Accepted Answer

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

Question 13

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 $1,000 if interest is paid at
A rate of 6% per year, compounded semiannually, for 7 years.
A)$617.78
B)$69.34
C)$661.12
D)$444.01
E)$665.06

Accepted Answer

The correct answer is C.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.The formula for the present value of a sum of money is:PV = FV / (1 + r/n)^(n*t)where:PV is the present valueFV is the future valuer is the annual interest raten is the number of times per year that interest is compoundedt is the number of yearsIn this case, we have:FV = $1,000r = 6% = 0.06n = 2 (since interest is compounded semiannually)t = 7Plugging these values into the formula, we get:PV = $1,000 / (1 + 0.06/2)^(2*7)PV = $1,000 / (1.03)^14PV = $661.12Therefore, the present value of $1,000 if interest is paid at a rate of 6% per year, compounded semiannually, for 7 years is $661.12.

Question 14

A cloth fragment is found in an ancient tomb.It contains 71% of the carbon-14 (half-life is 5700 years) that it is assumed to have had initially.How old is the cloth?
A)11,800 years
B)none of these
C)13,500 years
D)11,500 years
E)11,700 years

Accepted Answer

The answer of A cloth fragment is found in an...

Question 15

Use a calculator to find the value to four decimal places. $4 ^ { \sqrt { 3 } }$
A) $6.0000$
B) $6.9282$
C) $2.3094$
D) $11.0357$
E) $\quad 9.0000$

Accepted Answer

The answer of Use a calculator to find the value...

Question 16

Simplify the expression. $$5 ^ { \sqrt { 10 } } \cdot 5 ^ { \sqrt { 10 } }$$
A) $\quad 25 \sqrt { 10 }$
B) $\quad 5 ^ { \sqrt { 20 } }$
C) $25 \sqrt { 20 }$
D) $10 ^ { \sqrt { 5 } }$
E) $\quad 10 \sqrt { 20 }$

Accepted Answer

The answer of Simplify the expression. \[5 ^ { \sqrt...

Question 17

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 6 hours?
A)0.2%
B)16.5%
C)28.7%
D)0.8%
E)3.6%
​

Accepted Answer

To find the percent of the drug remaining after 6 hours, substitute $t = 6$ into the given equation: $P = e^{-0.3 \times 6} = e^{-1.8}$. Calculating $e^{-1.8}$ gives approximately 0.165, or 16.5% when converted to a percentage.

Question 18

Find the $\mathrm { pH }$ of a solution with a hydrogen ion concentration of $1.93 \cdot 10 ^ { - 4 }$ gram-ions per liter.
A) $\quad 3.7$
B) $\quad 0.372$
C) $\quad 3.72$
D) $\quad 3.63$
E) $\quad 3.67$

Accepted Answer

The answer of Find the $\mathrm { pH }$ of...

Question 19

A bacterial culture grows according to the formula $P = P _ { \mathrm { o } } a ^ { t }$. If it takes 4 days for the culture to triple in size, how long will it take to double in size?
A) $\quad - 0.9$ days
B) $\quad 2.1$ days
C) $\quad 0.4$ days
D) $\quad 5.0$ days
E) none of these

Accepted Answer

We know that after 4 days, the population has tripled. Therefore:

$$3P_{o} = P_{o}a^{4}$$

Simplifying, we get:

$$a = 3^{1/4}$$

To find out how long it takes for the population to double, we need to find the value of t that satisfies the equation:

$$2P_{o} = P_{o}(3^{1/4})^{t}$$

Simplifying, we get:

$$2 = 3^{t/4}$$

Taking the ln of both sides yields:

$$\ln(2) = \frac{t}{4}\ln(3)$$

Solving for t, we get:

$$t = \frac{4\ln(2)}{\ln(3)} \approx 0.4$$

Therefore, it takes approximately 0.4 days for the culture to double in size.

Question 20

A Wisconsin lake is stocked with 10,500 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) $\quad 42,000$ bluegill
B) $\quad 13,229$ bluegill
C) $\quad 63,000$ bluegill
D) $\quad 84,000$ bluegill
E) $\quad 126,000$ bluegill

Accepted Answer

The population doubles every 2 years. In 6 years, it will double 3 times: $10,500 \times 2^3 = 10,500 \times 8 = 84,000$.

Quiz 5: Exponential and Logarithmic Functions

Solve the equation. $\log ( x - 46 ) - \log 6 = \log ( x - 32 ) - \log x$

True or False? $\log _ { a } 7 = \log _ { 7 } a$

A bank credit card charges interest at the rate of 16% per year, compounded monthly.If a senior in college charges her last tuition bill of $1,700 and intends to pay it in one year, what will she have to Pay?

Solve the equation. $\log _ { 5 } x = \log _ { 5 } \frac { 1 } { x } + 8$

In one individual, the percent alcohol level $t$ minutes after two shots of whiskey is given by $P = 0.3 ( 1 -$ $e ^ { - 0.05 t }$ ) . Find the blood alcohol level after 29 minutes.

$\text { Tell whether the statement } \log _ { 7 } 7 ^ { 6 } = 6 \left( 7 ^ { \log 7 ^ { 6 } } \right) \text { is true or false. }$

Simplify the expression. $5 ^ { \log 57 }$

Solve the equation. $2 \log _ { 2 } x = 1 + \log _ { 2 } ( x + 40 )$

The concentration $x$ of a certain drug in an organ after $t$ minutes is given by $x = 0.08 \left( 1 - e ^ { - 0.1 t } \right)$ . Find the initial concentration of the drug when $t = 0$ .

An account now contains $11,300 and has been accumulating interest at a 9% annual rate, compounded continuously, for 9 years.Find the initial deposit.

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 $1,000 if interest is paid at A rate of 6% per year, compounded semiannually, for 7 years.

A cloth fragment is found in an ancient tomb.It contains 71% of the carbon-14 (half-life is 5700 years) that it is assumed to have had initially.How old is the cloth?

Use a calculator to find the value to four decimal places. $4 ^ { \sqrt { 3 } }$

Simplify the expression. $5 ^ { \sqrt { 10 } } \cdot 5 ^ { \sqrt { 10 } }$

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 6 hours?

Find the $\mathrm { pH }$ of a solution with a hydrogen ion concentration of $1.93 \cdot 10 ^ { - 4 }$ gram-ions per liter.

A bacterial culture grows according to the formula $P = P _ { \mathrm { o } } a ^ { t }$ . If it takes 4 days for the culture to triple in size, how long will it take to double in size?

A Wisconsin lake is stocked with 10,500 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?

Quiz 5: Exponential and Logarithmic Functions

Solve the equation. log⁡(x−46) −log⁡6=log⁡(x−32) −log⁡x\log ( x - 46 ) - \log 6 = \log ( x - 32 ) - \log xlog(x−46) −log6=log(x−32) −logx

True or False? log⁡a7=log⁡7a\log _ { a } 7 = \log _ { 7 } aloga​7=log7​a

A bank credit card charges interest at the rate of 16% per year, compounded monthly.If a senior in college charges her last tuition bill of $1,700 and intends to pay it in one year, what will she have to Pay?

Solve the equation. log⁡5x=log⁡51x+8\log _ { 5 } x = \log _ { 5 } \frac { 1 } { x } + 8log5​x=log5​x1​+8

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−0.05te ^ { - 0.05 t }e−0.05t ) . Find the blood alcohol level after 29 minutes.

Tell whether the statement log⁡776=6(7log⁡76) is true or false. \text { Tell whether the statement } \log _ { 7 } 7 ^ { 6 } = 6 \left( 7 ^ { \log 7 ^ { 6 } } \right) \text { is true or false. } Tell whether the statement log7​76=6(7log76) is true or false.

Simplify the expression. 5log⁡575 ^ { \log 57 }5log57

Solve the equation. 2log⁡2x=1+log⁡2(x+40) 2 \log _ { 2 } x = 1 + \log _ { 2 } ( x + 40 ) 2log2​x=1+log2​(x+40)

The concentration xxx of a certain drug in an organ after ttt minutes is given by x=0.08(1−e−0.1t) x = 0.08 \left( 1 - e ^ { - 0.1 t } \right) x=0.08(1−e−0.1t) . Find the initial concentration of the drug when t=0t = 0t=0 .