# Logarithms Questions and Answers

Math

LogarithmsLet f x log3 x 3 and g x log3 x 1 Part A If h x f x g x solve for h x in simplest form 4 points 4 Part B Determine the solution to the system of nonlinear equations 6 points

Math

LogarithmsSelect the correct answer from each drop down menu Tests show that the hydrogen ion concentration of a sample of apple juice is 0 0003 and that of ammonia is 1 3 x 10 9 Find the approximate pH of each liquid using the formula PH log H where H is the hydrogen ion concentration The pH value of the apple juice is The pH value of ammonia is J S O 54 V 21

Math

LogarithmsThe rabbit population of Springfield Ohio was 144 000 in 2016 It is expected to decrease by 7 2 per ye Write an exponential function and use it to predict the rabbit population of Springfield in 2036 Rabbit population of Springfield in 2036 Round answer to the nearest whole number

Math

LogarithmsIs x 0 a valid potential solution to the equation log x 2 log x 5 1 You don t need to solve this equation just determine if x 0 is a valid solution It can t be determined from the information given Yes since log 2 and log 5 both exist as valid values for the logarithm No because the logarithm of a negative value is undefined No because log 0 is undefined

Math

LogarithmsA student was asked to determine the y intercept for the logarithmic function f x log3 x 2 1 Which of the following expressions would result in the correct y intercept log2 1 log3 log3 log2 1 3 1 2 1 2

Math

LogarithmsRecall that the properties of logarithms also apply to the natural logarithms.
a. Rewrite 3In(x)+4ln(2)-2ln(y) using a single natural logarithm.
b. Rewrite In[(x - 3)(3x + 2)]³ in expanded form.

Math

LogarithmsAnesthesia. When a person has a cavity filled, the dentist typically gives a local anesthetic. After leaving the dentist's office, one's mouth often is numb for several more hours. If 70 ml of anesthesia is injected into the local tissue at the time of the procedure (t = 0), and the amount of anesthesia still in the local tissue / hours after the initial injection is given by A 70e-05, how much is in the local tissue 4 hours later?

Math

LogarithmsThe initial bacterium count in a culture is 1200. A biologist later makes a sample count of bacteria in
the culture and finds that the relative rate of growth is 15% per hour.
a. Find a function that models the number of bacteria after t hours. [ n(t)=n_e"]
b. What is the estimated count after 12 hours?
c. After how many hours will the bacteria count reach 500,000?

Math

LogarithmsSuppose a sample of a certain substance decayed to 69.8% of its original amount after 300 days. (Round your answers to two decimal places.)
(a) What is the half-life (in days) of this substance? days
(b) How long would it take the sample to decay to one-third of its original amount? days

Math

LogarithmsWarfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patient's body decreases at a rate proportional to the quantity remaining. Suppose that the half-life of warfarin in the body is 37 hours. Write a differential equation satisfied by Q: How many days does it take for the drug level in the body to be reduced to 35 percent of its original level? time =

Math

LogarithmsGiven f(x) = -log3 (x + 2) + 1, find the following.
a. Use the transformations to find three points on the graph and the equation of the asymptote.
b. Sketch the graph of f(x).

Math

LogarithmsHere are three problems.
(x-1)³-4 = 24
5^(x-1)-4 = 24
5^(x-1) — 4 = 24
Only one of these problems requires logarithms to solve. Which one? Explain why the other
two problems do not require logarithms to solve. Please explain your justification with as much
mathematical detail as possible.

Math

LogarithmsSolve the following logarithmic equation.
log (4x+6)= 1+ log (x - 5)
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is {}.
(Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.
B. There is no solution.

Math

LogarithmsUse the properties of logarithms to write the expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive.
log9(x+5)+ log9(x+3)
log9(x + 5) + log9(x+3)= ________

Math

LogarithmsSolve for x:
log6 (3x + 10) = 2
Write log6 (3x + 10) = 2 in exponential form.
_________________
Find the value of x.
x= ___________

Math

Logarithmse^(2x+1)=48
Write the equation as an equation involving logarithms.
_______________
Find the solution of the equation in terms of logarithms, or correct to four decimal places.
x=______________

Math

LogarithmsWhich is the logarithmic form of the given equation?
10ˣ = 1000
○ log 10 = x
○ log x = 10
○ log 1000 = x
○ log x = 1000

Math

LogarithmsGraph f(x)=-2 log3 (x) + 5 using transformations by first graphing the basic
logarithmic function (parent function), then the progression of transformations (in the
correct order). State the domain, range, and vertical asymptote and end behavior.

Math

LogarithmsUse the Change of Base Formula to evaluate log3 48. Then convert log3 48 to a logarithm in base 5. Round to the nearest thousandth.
3.524; logs 290.385
3.524; logs 80
3.871; logs 290.385
1.681; log, 80

Math

LogarithmsWhat is the domain and range of this function?
f(x) = logg(x-2) + 1
A domain: x > -2; range: all positive real numbers
B domain: x> 2; range: all positive real numbers
C domain: x>-2; range: all real numbers
D domain: x > 2; range: all real numbers

Math - Others

LogarithmsExpand the following expression: ln r³/x²y⁵
a. 3 In (r) + 2 In (x) -5 In (y)
b. 3 In (r) - 2 In (x) - 5 In (y)
c. 3 In (r) - 2 In (x) + 5 In (y)
d. 3 In (r) + 2 In (x) + 5 In (y)

Math

LogarithmsThe profits for a company can be modeled by the logarithmic function
P(t) = (3/2)log(2t + 1) With P in thousands of dollars and tin years. What is the average increase in profit from year 2 to 3?
a) 219 dollars/year
b) 0.23 dollars/year
c) 4.39 dollars/year
d) 4386 dollars/year

Math

LogarithmsHow much more intense is an earthquake of magnitude 3.6 than an earthquake of magnitude 1.8?
a) 63 times more intense
b) twice as intense
c) 100 times more intense
d) 1.8 times more intense

Math

LogarithmsWhich statement is TRUE for the graph of the logarithmic function f(x) = log₂x:
a) There is a horizontal asymptote
b) The graph only exists above the x-axis
c) The graph only exists to the right of the y-axis
d) There is no restriction on the domain

Math

LogarithmsExpand the logarithmic expression.
log₃ x⁶/5y
6log₃x - log₃5 + log₃y
6log₃x - log₃5-log₃ y
6log₃ x + log₃ 5 + log₃y
6log₃x - 5log₃y

Math

LogarithmsCondense the logarithmic expression.
log₅6+3log₅3-log₅9
a log₅153
b log₅24
c log₅6
d log₅18

Math

LogarithmsThe magnitude of an earthquake is measured on the Richter scale as a logarithm of the intensity of the shock wave. For magnitude R and intensity I, the formula is R = log(1). The September 20, 1999 earthquake in Taiwan measured 7.7 on the Richter scale. The Beverly Hills earthquake in January 23, 2002 measured 1.8 on the scale. How many times more intense was the Talwan earthquake than the Beverly Hills earthquake? Round your answer to two decimal places, if necessary.

Math

LogarithmsWhat is the solution to this equation?
log(2x 100) = 3
A. x= 100
B. x= 550
C. x= 1,000
D. x= 450

Math

LogarithmsLet a = log2 3, b= log2 5 and c=log₂ 7,
A. Write the following in terms of a, b and c
i. (K:2) log₂ (45): 2a+b-c
ii. (K:2) log₂ (52.5): a+b+c-1
B. (1:2) A student write the answer as log₂ (X) = 2a-3b+c-1, find the value of X:
(write the number in fraction, if write as 7/8)
C. (C:2) Assume that a positive integer Q is between 1 and 20. Find all possible values of Q such that
log₂ Q cannot be expressed using a, b and c (such as 1, 2, 3, 4, etc):
The reason is because

Math

LogarithmsUse a calculator to find the following natural logarithm.
In 67
In 67=
(Round to four decimal places as needed.)

Math

LogarithmsUse the Quotient Rule of Logarithms to write an expanded expression equivalent to log2(4x+1/x). Make sure to use parenthesis around your logarithm functions log(x + y).
Note: If you are using log you need to type it in and use the subscript button on the keyboard. There is no log button.
Provide your answer below:

Math

LogarithmsThere was a sample of 700 milligrams of a radioactive substance to start a study. Since then, the sample has decayed by 3.1% each year.
Let t be the number of years since the start of the study. Let y be the mass of the sample in milligrams.
Write an exponential function showing the relationship between y and f.

Math

LogarithmsUse the Product Rule of Logarithms to write the completely expanded expression equivalent to log(8(1 + 3x)). Make sure to use parenthesis around your logarithm functions log(x + y).

Math

LogarithmsWrite the following as the sum and/or difference of logarithms. Assume all variables are positive.
log (10z/4)= __
The answer format in lowercase characters is: log(number)
Spaces in the answer are optional.

Math

LogarithmsThe pH scale was designed to make it convenient to express hydrogen ion concentrations that are small in aqueous solutions. The definiton of pH is in terms of base 10 logarithms.
pH = -log[H+]
where [H+] is the hydrogen ion concentration.
a. If the hydrogen ion concentration in a solution is 7.60 × 10–5 mol/L, the pH is
b. If the pH of a solution is 3.165, the hydrogen ion concentration is
mol/L.

Math

LogarithmsUse the power property to rewrite each expression. Assume all variables are positive.
log, (a-4)=
The answer format in lowercase characters is: log_base (number)
Spaces in the answer are optional.

Math

LogarithmsFind the solution of the exponential equation
e²x+1= 32
in terms of logarithms, or correct to four decimal places.
x=

Math

LogarithmsWhich expression and justification completes the third step of her proof?
(A) log, (b^(xy)); power rule of exponents
(B) log,(b^(x-y)); subtraction property of exponents
(C) log(b^(x+y)); power rule of exponents
(D) log, (b^(x/y)), division property of exponents

Math

LogarithmsA student solved the equation below by graphing.
log6(x-1)= log₂(2x+2)
Which statement about the graph is true?
The curves do not intersect.
The curves intersect at one point.
The curves intersect at two points.
The curves appear to coincide.

Math

LogarithmsFind the equation of the function that is finally graphed after the following transformations are applied to the graph of y = 3ˣ (write your resulting transformation in every step as well as your final result):
a) Shift down 2 units
b) Shift the result of a) right 3 units
c) Reflect the result of b) about x-axis
d) Reflect the result of c) about y-axis

Math

LogarithmsWrite the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. Assume that the variables represent positive real numbers.
log₅ (x³/yz).

Math

LogarithmsWrite the expression as a sum and/or a difference of logarithms with all variables to the first degree.
log [9w(v + 4)]
(A) log 9+ log w+ log v + log 4
(B) log 9+ log w+ log (v + 4)
(C) log 9- log w-log (v + 4)
(D) 9log w+ log (v + 4)
(E) log (9w) + log (9v) + log 36

Math

LogarithmsThe relationship between water content (w%) and number of blows (N) in soils, as obtained from Casagrande's liquid limit device, is given by
w = 20-log10 N
The liquid limit of soil is
(a) 15.6%
(b) 16.6%
(c) 17.6%
(d) 18.6%