Question 1

Evaluate the expression. $\ln \left( e ^ { n + 1 } \right)$

Accepted Answer

A)  $- 1$ 
B)  $\pi + 1$ 
C)  $e ^ { n }$ 
D)  $\pi$ 
E)  $\ln ( x + 1 )$

Question 2

A wooden mast salvaged from a shipwreck contains $95.5 \%$ of the carbon-14 that is present in living trees. How long ago was the ship built? (The half-life of carbon-14 is $5730$ years.)

Accepted Answer

The answer of A wooden mast salvaged from a shipwreck...

Question 3

Evaluate the expression. $\log _ {5 } 625 - \log _ { 5 } 125$

Accepted Answer

A)  $1$ 
B)  $2$ 
C)  $3$ 
D)  $- 3$ 
E)  $- 1$

Question 4

Graph the function $y = 2 ^ { | x | }$ , not by plotting points but by applying your knowledge of the general shape of graphs of the form $a ^ { x }$ . State the domain, range, and asymptote of the function.

Accepted Answer

The answer of Graph the function \(y = 2 ^...

Question 5

Use the change of base formula and a calculator to evaluate $\log _ { 91 } 19$ correct to six decimal places.

Accepted Answer

The answer of Use the change of base formula and...

Question 6

A culture starts with $680$ bacteria. After just $30$ minutes the count is $3250$ . Find a formula for the number of bacteria after $t$ hours. Find the number of bacteria after $3.5$ hours. After how many minutes will the number of bacteria triple?

Accepted Answer

The answer of A culture starts with $680$ bacteria. After...

Question 7

Use a calculator to evaluate the expression $\log 0.5 \text {, }$ to four decimal places.

Accepted Answer

A)  $- 0.2241$ 
B)  $- 0.1042$ 
C)  $- 0.0314$ 
D)  $- 0.9421$ 
E)  $- 0.3010$

Question 8

Use a graphing calculator to find all solutions of the equation $\log _ {10 } x = 2 x ^ { 2 } - 5$ correct to two decimal places.

Accepted Answer

The answer of Use a graphing calculator to find all...

Question 9

Use the Laws of Logarithms to rewrite the expression $\ln \frac { e ^ { 3 x } \sqrt { x ^ { 2 } + 1 } } { x ^ { 3 } + y ^ { 3 } }$ in a form with no logarithm of a product, quotient, root, or powers.

Accepted Answer

The answer of Use the Laws of Logarithms to rewrite...

Question 10

For the function $y = \frac { 2 ^ { 2 / x } } { x }$ , ( $x < 0$ ), use a graphing calculator to find, correct to two decimal places, 
(a) the intervals on which the function is increasing or decreasing, and
(b) the range of the function.

Accepted Answer

The answer of For the function \(y = \frac {...

Question 11

How long will it take an investment of $\$ 300$ to triple, if the interest rate is $3.5 \%$ per year compounded continuously?

Accepted Answer

The answer of How long will it take an investment...

Question 12

A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function $m ( t ) = 13 e ^ { - 0.015 t }$ where $m ( t )$ is measured in kilograms. Round your answers to three decimal places.
a) Find the mass at time $t = 0$ .
b) How much of the mass remains after 50 days?

Accepted Answer

The answer of A radioactive substance decays in such a...

Question 13

If $10,000 is invested in an account at 5% per year, find the amount of the investment at the end of 12 years for the following compounding methods.
(a) Continuously
(b) Daily

Accepted Answer

The answer of If $10,000 is invested in an account...

Question 14

The count in a culture of bacteria was $867$ after $3$ hours and $906$ after $4$ hours.
(a) What is the relative rate of growth of the bacterial population? Express your answer as a percentage.
(b) What was the initial size of the culture?
(c) After how many hours with the bacteria reach $1200$ ?

Accepted Answer

The answer of The count in a culture of bacteria...

Question 15

Evaluate the expression. $\log _ { 7 } 1$

Accepted Answer

A)  $- 1$ 
B)  $1$ 
C)  $3$ 
D)  $5$ 
E)  $0$

Question 16

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by $t = 3 \frac { \log ( N / 50 ) } { \log 2 }$ Find the time required for the colony to grow to two million bacteria.

Accepted Answer

The answer of A certain strain of bacteria divides every...

Question 17

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $h ( x ) = e ^ { - x } , \quad h ( 1 ) , h ( \sqrt { 2 } ) , h ( - 3 ) , h ( 1 / 2 )$

Accepted Answer

The answer of Use a calculator to evaluate the function...

Question 18

Find the solution of the equation $3 ^ { x + 2 } = 5 ^ { 2 x }$ correct to four decimal places.

Accepted Answer

A)  $- 0.0365$ 
B)  $0.0265$ 
C)  $0.3785$ 
D)  $0.3932$ 
E)  $1.0363$

Question 19

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $g ( x ) = ( 2 / 3 ) ^ { 2 x } ; g ( 0.7 ) , g ( \sqrt { 2 } / 2 ) , g ( 1 / \pi ) , g ( 1 / 3 )$

Accepted Answer

The answer of Use a calculator to evaluate the function...

Question 20

Sketch the graph of the function $h ( x ) = ( 0.6 ) ^ { x }$ by making a table of values. Use a calculator if necessary.

Accepted Answer

The answer of Sketch the graph of the function \(h...

Exam 4: Exponential and Logarithmic Functions

Evaluate the expression. ln⁡(en+1) \ln \left( e ^ { n + 1 } \right) ln(en+1)

A wooden mast salvaged from a shipwreck contains 95.5%95.5 \%95.5% of the carbon-14 that is present in living trees. How long ago was the ship built? (The half-life of carbon-14 is 573057305730 years.)

Evaluate the expression. log⁡5625−log⁡5125\log _ {5 } 625 - \log _ { 5 } 125log5​625−log5​125

Graph the function y=2∣x∣y = 2 ^ { | x | }y=2∣x∣ , not by plotting points but by applying your knowledge of the general shape of graphs of the form axa ^ { x }ax . State the domain, range, and asymptote of the function.

Use the change of base formula and a calculator to evaluate log⁡9119\log _ { 91 } 19log91​19 correct to six decimal places.

A culture starts with 680680680 bacteria. After just 303030 minutes the count is 325032503250 . Find a formula for the number of bacteria after ttt hours. Find the number of bacteria after 3.53.53.5 hours. After how many minutes will the number of bacteria triple?

Use a calculator to evaluate the expression log⁡0.5, \log 0.5 \text {, }log0.5, to four decimal places.

Use a graphing calculator to find all solutions of the equation log⁡10x=2x2−5\log _ {10 } x = 2 x ^ { 2 } - 5log10​x=2x2−5 correct to two decimal places.

Use the Laws of Logarithms to rewrite the expression ln⁡e3xx2+1x3+y3\ln \frac { e ^ { 3 x } \sqrt { x ^ { 2 } + 1 } } { x ^ { 3 } + y ^ { 3 } }lnx3+y3e3xx2+1​​ in a form with no logarithm of a product, quotient, root, or powers.

For the function y=22/xxy = \frac { 2 ^ { 2 / x } } { x }y=x22/x​ , ( x<0x < 0x<0 ), use a graphing calculator to find, correct to two decimal places, (a) the intervals on which the function is increasing or decreasing, and (b) the range of the function.

How long will it take an investment of $300\$ 300$300 to triple, if the interest rate is 3.5%3.5 \%3.5% per year compounded continuously?

If $10,000 is invested in an account at 5% per year, find the amount of the investment at the end of 12 years for the following compounding methods. (a) Continuously (b) Daily

Evaluate the expression. log⁡71\log _ { 7 } 1log7​1

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. h(x)=e−x,h(1),h(2),h(−3),h(1/2)h ( x ) = e ^ { - x } , \quad h ( 1 ) , h ( \sqrt { 2 } ) , h ( - 3 ) , h ( 1 / 2 )h(x)=e−x,h(1),h(2​),h(−3),h(1/2)

Find the solution of the equation 3x+2=52x3 ^ { x + 2 } = 5 ^ { 2 x }3x+2=52x correct to four decimal places.

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. g(x)=(2/3)2x;g(0.7),g(2/2),g(1/π),g(1/3)g ( x ) = ( 2 / 3 ) ^ { 2 x } ; g ( 0.7 ) , g ( \sqrt { 2 } / 2 ) , g ( 1 / \pi ) , g ( 1 / 3 )g(x)=(2/3)2x;g(0.7),g(2​/2),g(1/π),g(1/3)

Sketch the graph of the function h(x)=(0.6)xh ( x ) = ( 0.6 ) ^ { x }h(x)=(0.6)x by making a table of values. Use a calculator if necessary.