Question 1

Write the exponential equation as an equation involving a logarithm.
-$$5 ^ { 3 } = x$$

Accepted Answer

The exponential equation $5^3 = x$ can be rewritten in logarithmic form as $\log_5 x = 3$, which means "the power to which 5 must be raised to get x is 3."

Question 2

Solve the problem.
-Which has a lower present value: $30,000 if interest is paid at a rate of 5.98% compounded continuously for 5 years, or $33,000 if interest is paid at a rate of 3.3% compounded continuously for 65 months?

Accepted Answer

To find the present value (PV) of each option, we use the formula for continuous compounding: $PV = Pe^{-rt}$, where $P$ is the future value, $r$ is the annual interest rate, and $t$ is the time in years. For the $30,000 at 5.98% for 5 years, $PV = 30000e^{-0.0598*5}$. For the $33,000 at 3.3% for 65 months (which is $65/12$ years), $PV = 33000e^{-0.033*(65/12)}$. Calculating these gives a lower present value for the $30,000 at 5.98% for 5 years, making option B correct.

Question 3

Write the exponential equation as an equation involving a logarithm.
-$$4 ^ { - 2 } = \frac { 1 } { 16 }$$

Accepted Answer

The exponential equation $4^{-2} = \frac{1}{16}$ can be rewritten in logarithmic form as $\log_{4}\frac{1}{16} = -2$, where the base of the logarithm (4) raised to the power on the right side of the equation (-2) equals the number inside the logarithm ($\frac{1}{16}$).

Question 4

Write the exponential equation as an equation involving a logarithm.
-$$4 ^ { 5 / 2 } = 32$$

Accepted Answer

To convert the exponential equation $4^{5/2} = 32$ to a logarithmic form, we use the definition of a logarithm: if $a^b = c$, then $\log_a c = b$. Applying this to the given equation, we get $\log_4 32 = \frac{5}{2}$, which matches choice B.

Question 5

Solve the problem.
-Instruments on a satellite measure the amount of power generated by the satellite's power supply. The time t and the power P can be modeled by the funct $$P = 50 e ^ { - t / 300 }$$ , where t is in days and P is in watts. How much power will be available after 378 days? Round to the nearest hundredth.

Accepted Answer

To find the power available after 378 days, substitute t = 378 into the function:  $$ P = 50 e^{-378/300} $$  $$ P = 50 e^{-1.26} $$  $$ P \approx 50 \times 0.283 $$  $$ P \approx 14.18 $$ watts.

Question 6

Evaluate the logarithm without the use of a calculator.
-$$\log _ { 9 } \frac { 1 } { 729 }$$

Accepted Answer

$\log_9 \frac{1}{729}$ can be rewritten as $\log_9 9^{-3}$ because $9^3 = 729$, and since the base is 9, this simplifies to $-3$.

Question 7

Write the logarithmic equation as an exponential equation.
-$\log _ { b } 36 = 2$

Accepted Answer

The logarithmic equation $\log_b 36 = 2$ can be rewritten as an exponential equation by raising the base $b$ to the power on the right side of the equation, which equals the number inside the log, resulting in $b^2 = 36$.

Question 8

Write the logarithmic equation as an exponential equation.
-$\log _ { 3 } 27 = x$

Accepted Answer

The logarithmic equation $\log _ { 3 } 27 = x$ means that 3 raised to the power of x equals 27, which is correctly represented by $3 ^ { x } = 27$.

Question 9

Solve the problem.
-An initial investment of $1740 is appreciated for 14 years in an account that earns 7% interest, compounded continuously. Find the amount of money in the account at the end of the period. Round to the nearest cent.

Accepted Answer

The formula for continuous compounding is A = Pe^(rt), where P is the principal amount ($1740), r is the interest rate (0.07), and t is the time in years (14). Plugging in these values, A = 1740 * e^(0.07*14) ≈ $4636.15.

Question 10

Write the exponential equation as an equation involving a logarithm.
-$$7 ^ { 3 } = 343$$

Accepted Answer

The correct logarithmic form of $7^3 = 343$ is $\log_7 343 = 3$, which means "to what power must 7 be raised to get 343?" The answer is 3.

Question 11

Solve the problem.
-Suppose your great-great grandmother invested $700 earning 3.6% interest compounded continuously 120 years ago. How much would her investment be worth today? Round to the nearest cent.

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested for in years. Plugging in the given values: $A = 700e^{0.036 \times 120}$, calculating this gives approximately $52,632.04.

Question 12

Evaluate the logarithm without the use of a calculator.
-$$\log _ { 4 } 16$$

Accepted Answer

The logarithm $\log _ { 4 } 16$ asks, "to what power must 4 be raised to get 16?" Since $4^2 = 16$, the answer is 2.

Question 13

Solve the problem.
-An original investment of $7000 earns 7% interest compounded continuously. What will the investment be worth in 3 years? Round to the nearest cent.

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested for in years. Plugging in the given values: $A = 7000e^{0.07*3}$, we find $A ≈ 8635.75$.

Question 14

Solve the problem.
-A rumor is spread at an elementary school with 1200 students according to the model $$\mathrm { N } = 1200 \left( 1 - \mathrm { e } ^ { - 0.16 \mathrm {~d} } \right)$$ where N is the number of students who have heard the rumor and d is the number of days that have elapsed since the rumor began. How many students will have heard the rumor after 5 days?

Accepted Answer

The answer of Solve the problem.
-A rumor is spread at...

Question 15

Write the logarithmic equation as an exponential equation.
-$\log _ { 5 } 25 = 2$

Accepted Answer

The logarithmic equation $\log _ { 5 } 25 = 2$ can be rewritten as an exponential equation where the base (5) raised to the power (2) equals the number (25), which matches option B: $5 ^ { 2 } = 25$.

Question 16

Solve the problem.
-Suppose that you have $12,000 to invest. Which investment yields the greater return over 7 years: 8.75% compounded continuously or 8.9% compounded semiannually?

Accepted Answer

The formula for continuously compounded interest is A = Pe^(rt), where A is the final amount, P is the principal, e is the base of the natural logarithm, r is the annual interest rate, and t is the time in years.The formula for semiannually compounded interest is A = P(1 + r/2)^(2t), where A is the final amount, P is the principal, r is the annual interest rate, and t is the time in years.Plugging in the values given in the problem, we get:For continuously compounded interest: A = 12000e^(0.0875*7) = 19,671.47For semiannually compounded interest: A = 12000(1 + 0.089/2)^(2*7) = 19,662.93Therefore, the investment that yields the greater return over 7 years is 8.75% compounded continuously.

Question 17

Write the exponential equation as an equation involving a logarithm.
-$$9 ^ { \mathrm { X } } = 81$$

Accepted Answer

To convert the exponential equation $9^x = 81$ to a logarithmic form, we use the definition of logarithms: if $a^b = c$, then $\log_a c = b$. Applying this to the given equation, we get $\log_9 81 = x$.

Question 18

Evaluate the logarithm without the use of a calculator.
-$$\log _ { 9 } \frac { 1 } { 81 }$$

Accepted Answer

The logarithm $\log _ { 9 } \frac { 1 } { 81 }$ can be rewritten as $\log _ { 9 } 9^{-2}$, since $81 = 9^2$ and using the property of exponents that $a^{-n} = \frac{1}{a^n}$. According to the logarithm rule $\log_a a^n = n$, the answer is $-2$.

Question 19

Write the logarithmic equation as an exponential equation.
-$\log _ { 2 } x = 3$

Accepted Answer

The logarithmic equation $\log _ { 2 } x = 3$ can be rewritten as an exponential equation where the base is 2, the exponent is 3, and the result is x, which matches option D: $2 ^ { 3 } = x$.

Question 20

Solve the problem.
-Suppose that $6000 is invested at an interest rate of 5.2% per year, compounded continuously. What is the balance after 2 years?Round to the nearest cent.

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested for in years. Plugging in the given values: $A = 6000e^{0.052 \times 2}$, we get $A \approx 6657.60$.

Quiz 5: Exponential and Logarithmic Functions and Equations

Write the exponential equation as an equation involving a logarithm. - $5 ^ { 3 } = x$

Solve the problem. -Which has a lower present value: $30,000 if interest is paid at a rate of 5.98% compounded continuously for 5 years, or $33,000 if interest is paid at a rate of 3.3% compounded continuously for 65 months?

Write the exponential equation as an equation involving a logarithm. - $4 ^ { - 2 } = \frac { 1 } { 16 }$

Write the exponential equation as an equation involving a logarithm. - $4 ^ { 5 / 2 } = 32$

Evaluate the logarithm without the use of a calculator. - $\log _ { 9 } \frac { 1 } { 729 }$

Write the logarithmic equation as an exponential equation. - $\log _ { b } 36 = 2$

Write the logarithmic equation as an exponential equation. - $\log _ { 3 } 27 = x$

Solve the problem. -An initial investment of $1740 is appreciated for 14 years in an account that earns 7% interest, compounded continuously. Find the amount of money in the account at the end of the period. Round to the nearest cent.

Write the exponential equation as an equation involving a logarithm. - $7 ^ { 3 } = 343$

Solve the problem. -Suppose your great-great grandmother invested $700 earning 3.6% interest compounded continuously 120 years ago. How much would her investment be worth today? Round to the nearest cent.

Evaluate the logarithm without the use of a calculator. - $\log _ { 4 } 16$

Solve the problem. -An original investment of $7000 earns 7% interest compounded continuously. What will the investment be worth in 3 years? Round to the nearest cent.

Write the logarithmic equation as an exponential equation. - $\log _ { 5 } 25 = 2$

Solve the problem. -Suppose that you have $12,000 to invest. Which investment yields the greater return over 7 years: 8.75% compounded continuously or 8.9% compounded semiannually?

Write the exponential equation as an equation involving a logarithm. - $9 ^ { \mathrm { X } } = 81$

Evaluate the logarithm without the use of a calculator. - $\log _ { 9 } \frac { 1 } { 81 }$

Write the logarithmic equation as an exponential equation. - $\log _ { 2 } x = 3$

Solve the problem. -Suppose that $6000 is invested at an interest rate of 5.2% per year, compounded continuously. What is the balance after 2 years?Round to the nearest cent.

Quiz 5: Exponential and Logarithmic Functions and Equations

Write the exponential equation as an equation involving a logarithm. - 53=x5 ^ { 3 } = x53=x

Solve the problem. -Which has a lower present value: $30,000 if interest is paid at a rate of 5.98% compounded continuously for 5 years, or $33,000 if interest is paid at a rate of 3.3% compounded continuously for 65 months?

Write the exponential equation as an equation involving a logarithm. - 4−2=1164 ^ { - 2 } = \frac { 1 } { 16 }4−2=161​

Write the exponential equation as an equation involving a logarithm. - 45/2=324 ^ { 5 / 2 } = 3245/2=32

Evaluate the logarithm without the use of a calculator. - log⁡91729\log _ { 9 } \frac { 1 } { 729 }log9​7291​

Write the logarithmic equation as an exponential equation. - log⁡b36=2\log _ { b } 36 = 2logb​36=2

Write the logarithmic equation as an exponential equation. - log⁡327=x\log _ { 3 } 27 = xlog3​27=x