Question 1

Use the definition of the logarithmic function to find $x$ if $\log _ { x } 512 = 3$ .

Accepted Answer

$\log _ { x } 512 = 3$ $\Leftrightarrow$ $x ^ { 3 } = 512$ $\Leftrightarrow$ $x ^ { 3 } = 8 ^ { 3 }$ $\Leftrightarrow$ $x = 8$

Question 2

Find the solution of the equation $2 ^ { x/2 } = 0.01$ correct to four decimal places.

Accepted Answer

A)  $- 53.1508$ 
B)  $- 29.0743$ 
C)  $- 25.1508$ 
D)  $- 19.0828$ 
E)  $- 7.0523$ 
A)  $- 53.1508$ 
B)  $- 29.0743$ 
C)  $- 25.1508$ 
D)  $- 19.0828$ 
E)  $- 7.0523$ A

Question 3

If the pH readings of an unknown substance is pH $= 10.4$ , find the hydrogen ion concentration.

Accepted Answer

pH $= - \log \left[ \mathrm { H } ^ { .} \right] = 10.4$ $\Leftrightarrow$ $\left[ \mathrm { H } ^ {.} \right] = 10 ^ { - 10.4 }$ M $\approx 4.0 \times 10 ^ { - 11 }$ M

Question 4

The population of a certain country was $2,345,235$ in $1985$ and $3,458,377$ in $1995 .$ Assume that the growth is exponential. 
(a) Find a formula for the population $t$ years after $1985 .$ 
(b) Find the time required for the population to double.
(c) Use the data to predict the population in the year $2005 .$

Accepted Answer

a) Since the initial time is @#LAT-DLM& , @#LAT-DLM& , @#LAT-DLM& , @#LAT-DLM& a

Question 5

Find the solution of the exponential equation $e ^ { [ ( x / 4 ) - 1 } = 15$ , correct to four decimal places.

Accepted Answer

The answer of Find the solution of the exponential equation...

Question 6

If $f ( x ) = 5 ^ { x }$ , show that $\frac { f ( x + h ) - f ( x ) } { h } = 5 ^ { x } \left( \frac { 5 ^ { h } - 1 } { h } \right)$ .

Accepted Answer

The answer of If \(f ( x ) = 5...

Question 7

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 100 days?

Accepted Answer

a) 13 kg b

Question 8

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If $D _ { 0 }$ is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by $A = - 8267 \ln \left( \frac { D } { D _ { 0 } } \right)$ Find the age of an object if the amount D of carbon-14 that remains in the object is 50% of the original amount $D _ { 0 }$ .

Accepted Answer

The answer of The age of an ancient artifact can...

Question 9

Show that $- \log ( \sqrt { x + 1 } - \sqrt { x } ) = \log ( \sqrt { x + 1 } + \sqrt { x } )$ .

Accepted Answer

None

Question 10

Solve the equation $2 x ^ { 2 } \left( 3 ^ { x } \right) - 5 x \left( 3 ^ { x } \right) + 3 ^ { x + 1 } = 0$ .

Accepted Answer

@#LAT-DLM&
. Now @#LAT-DLM& has no solut

Question 11

Use the Laws of Logarithms to rewrite the expression $\log _ { 4 } \left( \frac { x ^ { 4 } } { 4 } \right)$ 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 12

If $10,000 is invested at an interest rate of 5% per year, compounded semiannually, find the value of the investment after the given number of years.
(a) 5 years
(b) 10 years
(c) 15 years

Accepted Answer

$12,800.86

Question 13

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If $D _ { 0 }$ is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by $A = - 8267 \ln \left( \frac { D } { D _ { 0 } } \right)$ Find the age of an object if the amount D of carbon-14 that remains in the object is 85% of the original amount $D _ { 0 }$ .

Accepted Answer

The answer of The age of an ancient artifact can...

Question 14

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 five million bacteria.

Accepted Answer

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

Question 15

Find the exponential function whose graph is given.

Accepted Answer

The answer of Find the exponential function whose graph is...

Question 16

Rudy wants to invest $\$ 1000$ in savings certificates that bear an interest rate of $7.5 \%$ compounded semiannually. How long will it take for the amount to be $\$ 1500 ?$

Accepted Answer

A)  $2.54$ yrs 
B)  $3.08$ yrs 
C)  $5.51$ yrs 
D)  $9.25$ yrs 
E)  $10.36$ yrs 
A)  $2.54$ yrs 
B)  $3.08$ yrs 
C)  $5.51$ yrs 
D)  $9.25$ yrs 
E)  $10.36$ yrs

Question 17

Find the effective yield for an investment that earns 3% per year, compounded quarterly.

Accepted Answer

After one year, p gr

Question 18

Use the Laws of Logarithms to rewrite $\log _ { 11 } \sqrt [ 3 ] { 10 }$ in a form with no logarithms of products, quotients, or powers.

Accepted Answer

A)  $\frac { 1 } { 11 } \log _ { 3 } 10$ 
B)  $\log _ { 11 } 10 - \log _ { 11 } 3$ 
C)  $\sqrt [ 3 ] { \log _ { 11 } 10 }$ 
D)  $\frac { 1 } { 3 } \log _ { 11 } 10$ 
E)  $\frac { 1 } { 3 } \log _ { 1 } 3$ 
A)  $\frac { 1 } { 11 } \log _ { 3 } 10$ 
B)  $\log _ { 11 } 10 - \log _ { 11 } 3$ 
C)  $\sqrt [ 3 ] { \log _ { 11 } 10 }$ 
D)  $\frac { 1 } { 3 } \log _ { 11 } 10$ 
E)  $\frac { 1 } { 3 } \log _ { 1 } 3$

Question 19

A kettle full of water is brought to a boil in a room with temperature $32 ^ { \circ }$ C. After $5$ minutes the temperature of the water has decreased from $100 ^ { \circ }$ C to $80 ^ { \circ }$ C. Find the temperature after another $5$ minutes.

Accepted Answer

@#LAT-DLM& and @#LAT-DLM& . Then @#LAT-DLM& . Sinc

Question 20

Use the change of base formula and a calculator to evaluate $\log _ { 6 } 3.58$ correct to six decimal places.

Accepted Answer

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

Use the definition of the logarithmic function to find $x$ if $\log _ { x } 512 = 3$ .

Find the solution of the equation $2 ^ { x/2 } = 0.01$ correct to four decimal places.

If the pH readings of an unknown substance is pH $= 10.4$ , find the hydrogen ion concentration.

Find the solution of the exponential equation $e ^ { [ ( x / 4 ) - 1 } = 15$ , correct to four decimal places.

If $f ( x ) = 5 ^ { x }$ , show that $\frac { f ( x + h ) - f ( x ) } { h } = 5 ^ { x } \left( \frac { 5 ^ { h } - 1 } { h } \right)$ .

Show that $- \log ( \sqrt { x + 1 } - \sqrt { x } ) = \log ( \sqrt { x + 1 } + \sqrt { x } )$ .

Solve the equation $2 x ^ { 2 } \left( 3 ^ { x } \right) - 5 x \left( 3 ^ { x } \right) + 3 ^ { x + 1 } = 0$ .

Use the Laws of Logarithms to rewrite the expression $\log _ { 4 } \left( \frac { x ^ { 4 } } { 4 } \right)$ in a form with no logarithm of a product, quotient, root, or powers.

If $10,000 is invested at an interest rate of 5% per year, compounded semiannually, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years

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 five million bacteria.

Find the exponential function whose graph is given.

Rudy wants to invest $\$ 1000$ in savings certificates that bear an interest rate of $7.5 \%$ compounded semiannually. How long will it take for the amount to be $\$ 1500 ?$

Find the effective yield for an investment that earns 3% per year, compounded quarterly.

Use the Laws of Logarithms to rewrite $\log _ { 11 } \sqrt [ 3 ] { 10 }$ in a form with no logarithms of products, quotients, or powers.

A kettle full of water is brought to a boil in a room with temperature $32 ^ { \circ }$ C. After $5$ minutes the temperature of the water has decreased from $100 ^ { \circ }$ C to $80 ^ { \circ }$ C. Find the temperature after another $5$ minutes.

Use the change of base formula and a calculator to evaluate $\log _ { 6 } 3.58$ correct to six decimal places.

Fundamentals

Functions

Polynomial and Rational Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Exam 4: Exponential and Logarithmic Functions

Use the definition of the logarithmic function to find xxx if log⁡x512=3\log _ { x } 512 = 3logx​512=3 .

Find the solution of the equation 2x/2=0.012 ^ { x/2 } = 0.012x/2=0.01 correct to four decimal places.

If the pH readings of an unknown substance is pH =10.4= 10.4=10.4 , find the hydrogen ion concentration.

Find the solution of the exponential equation e[(x/4)−1=15e ^ { [ ( x / 4 ) - 1 } = 15e[(x/4)−1=15 , correct to four decimal places.

If f(x)=5xf ( x ) = 5 ^ { x }f(x)=5x , show that f(x+h)−f(x)h=5x(5h−1h)\frac { f ( x + h ) - f ( x ) } { h } = 5 ^ { x } \left( \frac { 5 ^ { h } - 1 } { h } \right)hf(x+h)−f(x)​=5x(h5h−1​) .

Show that −log⁡(x+1−x)=log⁡(x+1+x)- \log ( \sqrt { x + 1 } - \sqrt { x } ) = \log ( \sqrt { x + 1 } + \sqrt { x } )−log(x+1​−x​)=log(x+1​+x​) .

Solve the equation 2x2(3x)−5x(3x)+3x+1=02 x ^ { 2 } \left( 3 ^ { x } \right) - 5 x \left( 3 ^ { x } \right) + 3 ^ { x + 1 } = 02x2(3x)−5x(3x)+3x+1=0 .

Use the Laws of Logarithms to rewrite the expression log⁡4(x44)\log _ { 4 } \left( \frac { x ^ { 4 } } { 4 } \right)log4​(4x4​) in a form with no logarithm of a product, quotient, root, or powers.

If $10,000 is invested at an interest rate of 5% per year, compounded semiannually, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years

Find the exponential function whose graph is given.

Rudy wants to invest $1000\$ 1000$1000 in savings certificates that bear an interest rate of 7.5%7.5 \%7.5% compounded semiannually. How long will it take for the amount to be $1500?\$ 1500 ?$1500?

Find the effective yield for an investment that earns 3% per year, compounded quarterly.

Use the Laws of Logarithms to rewrite log⁡11103\log _ { 11 } \sqrt [ 3 ] { 10 }log11​310​ in a form with no logarithms of products, quotients, or powers.

A kettle full of water is brought to a boil in a room with temperature 32∘32 ^ { \circ }32∘ C. After 555 minutes the temperature of the water has decreased from 100∘100 ^ { \circ }100∘ C to 80∘80 ^ { \circ }80∘ C. Find the temperature after another 555 minutes.

Use the change of base formula and a calculator to evaluate log⁡63.58\log _ { 6 } 3.58log6​3.58 correct to six decimal places.

Fundamentals

Functions

Polynomial and Rational Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Use the definition of the logarithmic function to find $x$ if $\log _ { x } 512 = 3$ .

Find the solution of the equation $2 ^ { x/2 } = 0.01$ correct to four decimal places.

If the pH readings of an unknown substance is pH $= 10.4$ , find the hydrogen ion concentration.

Find the solution of the exponential equation $e ^ { [ ( x / 4 ) - 1 } = 15$ , correct to four decimal places.

If $f ( x ) = 5 ^ { x }$ , show that $\frac { f ( x + h ) - f ( x ) } { h } = 5 ^ { x } \left( \frac { 5 ^ { h } - 1 } { h } \right)$ .

Show that $- \log ( \sqrt { x + 1 } - \sqrt { x } ) = \log ( \sqrt { x + 1 } + \sqrt { x } )$ .

Solve the equation $2 x ^ { 2 } \left( 3 ^ { x } \right) - 5 x \left( 3 ^ { x } \right) + 3 ^ { x + 1 } = 0$ .

Use the Laws of Logarithms to rewrite the expression $\log _ { 4 } \left( \frac { x ^ { 4 } } { 4 } \right)$ in a form with no logarithm of a product, quotient, root, or powers.

Rudy wants to invest $\$ 1000$ in savings certificates that bear an interest rate of $7.5 \%$ compounded semiannually. How long will it take for the amount to be $\$ 1500 ?$

Use the Laws of Logarithms to rewrite $\log _ { 11 } \sqrt [ 3 ] { 10 }$ in a form with no logarithms of products, quotients, or powers.

A kettle full of water is brought to a boil in a room with temperature $32 ^ { \circ }$ C. After $5$ minutes the temperature of the water has decreased from $100 ^ { \circ }$ C to $80 ^ { \circ }$ C. Find the temperature after another $5$ minutes.

Use the change of base formula and a calculator to evaluate $\log _ { 6 } 3.58$ correct to six decimal places.