Question 1

Provide an appropriate response.
-$\text { Graph } y = \left( \frac { 1 } { 4 } \right) ^ { x } + 2$

Accepted Answer

The answer of Provide an appropriate response.
-\(\text { Graph }...

Question 2

Provide an appropriate response.
-Graph the functions $y = 3 ^ { x }$ and $y = \log _ { 3 } x$ on the same coordinate system.

Accepted Answer

The answer of Provide an appropriate response.
-Graph the functions \(y...

Question 3

Use the properties of logarithms to write the expression as a single logarithm.
-$\log _ { 6 } x + 4 \log _ { 6 } x - \log _ { 6 } ( x + 9 )$

Accepted Answer

Using the properties of logarithms, the expression can be simplified as follows: $\log_6 x + 4\log_6 x = 5\log_6 x$, which can be written as $\log_6 x^5$. Then, subtracting $\log_6(x + 9)$ gives $\log_6 \frac{x^5}{x + 9}$.

Question 4

Use the formula A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }$
-$\$ 3500$ is invested at $5 \%$ compounded quarterly. In how many years will the account have grown to $\$ 8000$ ? Round to the nearest tenth of a year.

Accepted Answer

To solve for $t$, we use the formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ with $A = 8000$, $P = 3500$, $r = 0.05$ (5% as a decimal), and $n = 4$ (compounded quarterly). Rearranging for $t$, we get $t = \frac{\log(A/P)}{n \log(1 + r/n)}$. Plugging in the values, $t$ calculates to approximately $16.6$ years.

Question 5

Provide an appropriate response.
-Write the expression $\log _ { 5 } \frac { 4 x ^ { 6 } } { y ^ { 7 } }$ as a sum or difference of multiples of logarithms.

Accepted Answer

Using the logarithm properties, the expression can be rewritten as a sum of the logarithm of the numerator and the negative of the logarithm of the denominator, and the exponent can be brought in front as a multiplier.

Question 6

Provide an appropriate response.
-Approximate $\log _ { 2 } 5$ to four decimal places.

Accepted Answer

To approximate $\log_2 5$, we can use a calculator or logarithmic tables. The base-2 logarithm of 5 is about 2.3219 when rounded to four decimal places.

Question 7

Solve the logarithmic equation for x. Give an exact solution
-$\log _ { 8 } ( 3 x - 2 ) = 2$

Accepted Answer

The answer of Solve the logarithmic equation for x. Give...

Question 8

Provide an appropriate response.
-If $\log _ { 10 } 2 = 0.3010$ and $\log _ { 10 } 3 = 0.4771$, find the value of $\log _ { 10 } \frac { 27 } { 4 }$.

Accepted Answer

Using the given logarithm values, $\log _ { 10 } \frac { 27 } { 4 } = \log _ { 10 } (3^3) - \log _ { 10 } (2^2) = 3\log _ { 10 } 3 - 2\log _ { 10 } 2 = 3(0.4771) - 2(0.3010) = 1.4313 - 0.6020 = 0.8293.$

Question 9

Solve. Round the answer to the nearest whole number.
-The size of the coyote population at a national park increases at the rate of $4.8 \%$ per year. If the size of the current population is 149 , find how many coyotes there will be in 7 years.

Accepted Answer

The future population can be calculated using the formula for compound interest: $ P = P_0 (1 + r)^t $. Here, $ P_0 = 149 $, $ r = 0.048 $, and $ t = 7 $. Calculating gives $ P \approx 207.37 $, which rounds to 207.

Question 10

Solve. Round the answer to the nearest whole number.
-If the annual population growth rate is $5.9 \%$, find how long will it take a population of birds to increase from 200 birds to 1100 birds.

Accepted Answer

To solve this problem, we can use the formula for exponential growth: $P = P_0 \times (1 + r)^t$, where $P$ is the future population, $P_0$ is the initial population, $r$ is the growth rate (as a decimal), and $t$ is the time in years. Plugging in the given values, we have $1100 = 200 \times (1 + 0.059)^t$. Solving for $t$, we get $t \approx 29$ years.

Question 11

Use the formula A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }$
-Suppose that you have $7000 to invest. Which investment yields the greater return over 6 years: 6.6% compounded monthly or 6.7% compounded quarterly?

Accepted Answer

Using the formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$, we calculate the future value for both options. For 6.6% compounded monthly, $n = 12$ and $r = 0.066$, so $A = 7000 \left( 1 + \frac { 0.066 } { 12 } \right) ^ { 12 \times 6 }$. For 6.7% compounded quarterly, $n = 4$ and $r = 0.067$, so $A = 7000 \left( 1 + \frac { 0.067 } { 4 } \right) ^ { 4 \times 6 }$. Calculating both, the 6.7% compounded quarterly yields a greater return.

Question 12

Provide an appropriate response.
-Solve $6 ^ { x - 2 } = \frac { 1 } { 36 }$ for $x$. Give an exact solution.

Accepted Answer

The answer of Provide an appropriate response.
-Solve \(6 ^ {...

Question 13

Solve the logarithmic equation for x. Give an exact solution
-$\log _ { 5 } x = - 3$

Accepted Answer

To solve the equation $\log_5 x = -3$, we rewrite it in exponential form: $5^{-3} = x$. This simplifies to $x = \frac{1}{5^3} = \frac{1}{125}$.

Question 14

Solve the logarithmic equation for x. Give an exact solution
-$\log _ { 5 } 3 + \log _ { 5 } x = 1$

Accepted Answer

Using the logarithm property $\log_b(m) + \log_b(n) = \log_b(mn)$, the equation becomes $\log_5(3x) = 1$. Solving for $x$ using the definition of a logarithm, $5^1 = 3x$, gives $x = \frac{5}{3}$.

Question 15

Provide an appropriate response.
-Solve $2^{2 x + 1 }= 7$ for $x$. Give an exact solution and a four-decimal-place approximation.

Accepted Answer

To solve $2^{2x + 1} = 7$, we first isolate the exponential expression: $2^{2x + 1} = 7$. Taking the logarithm of both sides gives $2x + 1 = \frac{\log 7}{\log 2}$. Solving for $x$ yields $x = \frac{1}{2} \left( \frac{\log 7}{\log 2} - 1 \right)$. The approximation of this value is 0.9037.

Question 16

Solve the logarithmic equation for x. Give an exact solution
-$\log _ { 3 } ( x + 2 ) - \log _ { 3 } ( x - 4 ) = 3$

Accepted Answer

Using the properties of logarithms, we can simplify the equation as follows:$\log _ { 3 } ( x + 2 ) - \log _ { 3 } ( x - 4 ) = 3$$\log _ { 3 } \left ( \frac { x + 2 } { x - 4 } \right ) = 3$$\frac { x + 2 } { x - 4 } = 3 ^ { 3 } = 27$$x + 2 = 27 ( x - 4 )$$x + 2 = 27 x - 108$$-26 x = -110$$x = \frac { 110 } { 26 } = \frac { 55 } { 13 }$

Question 17

Provide an appropriate response.
-Solve $\ln ( 7 x + 1 ) = 1.56$ accurate to four decimal places.

Accepted Answer

To solve the equation $\ln(7x + 1) = 1.56$, first exponentiate both sides to get rid of the natural logarithm: $e^{\ln(7x + 1)} = e^{1.56}$. This simplifies to $7x + 1 = e^{1.56}$. Solving for $x$ gives $x = \frac{e^{1.56} - 1}{7}$. Using a calculator, this gives $x \approx 0.5370$.

Question 18

Use the formula A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }$
-Find the amount of money in an account after 3 years if $2300 is deposited at 8% annual interest compounded monthly.

Accepted Answer

The answer of Use the formula A \(A = P...

Question 19

Solve. Round the answer to the nearest whole number.
-Suppose a city with population of 170,000 has been decreasing at a rate of 2% per year. If this rate continues, predict the population of the city in 10 years.

Accepted Answer

To predict the population of the city in 10 years with a 2% annual decrease, we use the formula for exponential decay: P = P0(1 - r)^t, where P0 is the initial population, r is the rate of decrease (as a decimal), and t is the time in years. Plugging in the values: P = 170,000(1 - 0.02)^10 ≈ 138,902 when rounded to the nearest whole number.

Question 20

Solve the logarithmic equation for x. Give an exact solution
-$\ln \sqrt [ 7 ] { e } = x$

Accepted Answer

The equation can be simplified using the properties of logarithms and exponents: $\ln(e^{1/7}) = x$. Since $\ln(e^y) = y$, the solution is $x = \frac{1}{7}$.

Quiz 12: Exponential and Logarithmic Functions

Provide an appropriate response. - $\text { Graph } y = \left( \frac { 1 } { 4 } \right) ^ { x } + 2$ $Provide an appropriate response. - \text { Graph } y = \left( \frac { 1 } { 4 } \right) ^ { x } + 2 A) B) C) D)$

Provide an appropriate response. -Graph the functions $y = 3 ^ { x }$ and $y = \log _ { 3 } x$ on the same coordinate system. $Provide an appropriate response. -Graph the functions y = 3 ^ { x } and y = \log _ { 3 } x on the same coordinate system. A) B) C) D)$

Use the properties of logarithms to write the expression as a single logarithm. - $\log _ { 6 } x + 4 \log _ { 6 } x - \log _ { 6 } ( x + 9 )$

Use the formula A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }$ - $\$ 3500$ is invested at $5 \%$ compounded quarterly. In how many years will the account have grown to $\$ 8000$ ? Round to the nearest tenth of a year.

Provide an appropriate response. -Write the expression $\log _ { 5 } \frac { 4 x ^ { 6 } } { y ^ { 7 } }$ as a sum or difference of multiples of logarithms.

Provide an appropriate response. -Approximate $\log _ { 2 } 5$ to four decimal places.

Solve the logarithmic equation for x. Give an exact solution - $\log _ { 8 } ( 3 x - 2 ) = 2$

Provide an appropriate response. -If $\log _ { 10 } 2 = 0.3010$ and $\log _ { 10 } 3 = 0.4771$ , find the value of $\log _ { 10 } \frac { 27 } { 4 }$ .

Solve. Round the answer to the nearest whole number. -The size of the coyote population at a national park increases at the rate of $4.8 \%$ per year. If the size of the current population is 149 , find how many coyotes there will be in 7 years.

Solve. Round the answer to the nearest whole number. -If the annual population growth rate is $5.9 \%$ , find how long will it take a population of birds to increase from 200 birds to 1100 birds.

Use the formula A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }$ -Suppose that you have $7000 to invest. Which investment yields the greater return over 6 years: 6.6% compounded monthly or 6.7% compounded quarterly?

Provide an appropriate response. -Solve $6 ^ { x - 2 } = \frac { 1 } { 36 }$ for $x$ . Give an exact solution.

Solve the logarithmic equation for x. Give an exact solution - $\log _ { 5 } x = - 3$

Solve the logarithmic equation for x. Give an exact solution - $\log _ { 5 } 3 + \log _ { 5 } x = 1$

Provide an appropriate response. -Solve $2^{2 x + 1 }= 7$ for $x$ . Give an exact solution and a four-decimal-place approximation.

Solve the logarithmic equation for x. Give an exact solution - $\log _ { 3 } ( x + 2 ) - \log _ { 3 } ( x - 4 ) = 3$

Provide an appropriate response. -Solve $\ln ( 7 x + 1 ) = 1.56$ accurate to four decimal places.

Use the formula A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }$ -Find the amount of money in an account after 3 years if $2300 is deposited at 8% annual interest compounded monthly.

Solve. Round the answer to the nearest whole number. -Suppose a city with population of 170,000 has been decreasing at a rate of 2% per year. If this rate continues, predict the population of the city in 10 years.

Solve the logarithmic equation for x. Give an exact solution - $\ln \sqrt [ 7 ] { e } = x$

Quiz 12: Exponential and Logarithmic Functions

Provide an appropriate response. - Graph y=(14) x+2\text { Graph } y = \left( \frac { 1 } { 4 } \right) ^ { x } + 2 Graph y=(41​) x+2

Provide an appropriate response. -Graph the functions y=3xy = 3 ^ { x }y=3x and y=log⁡3xy = \log _ { 3 } xy=log3​x on the same coordinate system.

Use the properties of logarithms to write the expression as a single logarithm. - log⁡6x+4log⁡6x−log⁡6(x+9) \log _ { 6 } x + 4 \log _ { 6 } x - \log _ { 6 } ( x + 9 ) log6​x+4log6​x−log6​(x+9)

Provide an appropriate response. -Write the expression log⁡54x6y7\log _ { 5 } \frac { 4 x ^ { 6 } } { y ^ { 7 } }log5​y74x6​ as a sum or difference of multiples of logarithms.

Provide an appropriate response. -Approximate log⁡25\log _ { 2 } 5log2​5 to four decimal places.

Solve the logarithmic equation for x. Give an exact solution - log⁡8(3x−2) =2\log _ { 8 } ( 3 x - 2 ) = 2log8​(3x−2) =2

Provide an appropriate response. -If log⁡102=0.3010\log _ { 10 } 2 = 0.3010log10​2=0.3010 and log⁡103=0.4771\log _ { 10 } 3 = 0.4771log10​3=0.4771 , find the value of log⁡10274\log _ { 10 } \frac { 27 } { 4 }log10​427​ .

Solve. Round the answer to the nearest whole number. -The size of the coyote population at a national park increases at the rate of 4.8%4.8 \%4.8% per year. If the size of the current population is 149 , find how many coyotes there will be in 7 years.

Solve. Round the answer to the nearest whole number. -If the annual population growth rate is 5.9%5.9 \%5.9% , find how long will it take a population of birds to increase from 200 birds to 1100 birds.

Use the formula A A=P(1+rn) nt to solve. A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }A=P(1+nr​) nt to solve. -Suppose that you have $7000 to invest. Which investment yields the greater return over 6 years: 6.6% compounded monthly or 6.7% compounded quarterly?

Provide an appropriate response. -Solve 6x−2=1366 ^ { x - 2 } = \frac { 1 } { 36 }6x−2=361​ for xxx . Give an exact solution.

Solve the logarithmic equation for x. Give an exact solution - log⁡5x=−3\log _ { 5 } x = - 3log5​x=−3

Solve the logarithmic equation for x. Give an exact solution - log⁡53+log⁡5x=1\log _ { 5 } 3 + \log _ { 5 } x = 1log5​3+log5​x=1

Provide an appropriate response. -Solve 22x+1=72^{2 x + 1 }= 722x+1=7 for xxx . Give an exact solution and a four-decimal-place approximation.

Solve the logarithmic equation for x. Give an exact solution - log⁡3(x+2) −log⁡3(x−4) =3\log _ { 3 } ( x + 2 ) - \log _ { 3 } ( x - 4 ) = 3log3​(x+2) −log3​(x−4) =3

Provide an appropriate response. -Solve ln⁡(7x+1) =1.56\ln ( 7 x + 1 ) = 1.56ln(7x+1) =1.56 accurate to four decimal places.

Use the formula A A=P(1+rn) nt to solve. A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { to solve. }A=P(1+nr​) nt to solve. -Find the amount of money in an account after 3 years if $2300 is deposited at 8% annual interest compounded monthly.