Lesson Worksheet: Logarithmic Functions Mathematics

In this worksheet, we will practice identifying, writing, and evaluating a logarithmic function as an inverse of the exponential function.

Q1:

The function 𝑓(𝑥)=2𝑒+3 has an inverse of the form 𝑔(𝑥)=(𝑎𝑥+𝑏)ln. What are the values of 𝑎 and 𝑏?

  • A𝑎=12, 𝑏=−32
  • B𝑎=−32, 𝑏=12
  • C𝑎=12, 𝑏=32
  • D𝑎=1, 𝑏=−3
  • E𝑎=2, 𝑏=3

Q2:

Make 𝑥 the subject of the equation 𝑦=2, assuming that 𝑎≠0.

  • A𝑥=(𝑦)−𝑏𝑎log
  • B𝑥=(𝑦)−𝑏log
  • C𝑥=(𝑦−2)−𝑏𝑎log
  • D𝑦=(𝑥)−𝑏𝑎log
  • E𝑥=(𝑦)+𝑏𝑎log

Q3:

Rearrange the equation 𝑦=2−12+1 to find 𝑥 in terms of 𝑦. Hence determine the inverse 𝑔 to the function 𝑔(𝑥)=2−12+1.

  • A𝑔(𝑦)=−𝑦−1𝑦−2log
  • B𝑔(𝑥)=−𝑥−1𝑥−2log
  • C𝑔(𝑥)=−𝑥−1𝑥−2log
  • D𝑔(𝑥)=𝑥+1𝑥−2log
  • E𝑔(𝑦)=−𝑦−1𝑦−2log

Q4:

The function 𝑓(𝑥)=(4𝑥)−1log has an inverse of the form 𝑔(𝑥)=𝐴3. Determine the values of 𝐴 and 𝑘.

  • A𝑘=1, 𝐴=14
  • B𝑘=1, 𝐴=34
  • C𝑘=3, 𝐴=4
  • D𝑘=34, 𝐴=1
  • E𝑘=−1, 𝐴=112

Q5:

Consider the function 𝑓(𝑥)=𝑥log. Find the value of 𝑓(2).

  • A14
  • B13
  • C−3
  • D4
  • E3

Q6:

Consider the function 𝑓(𝑥)=𝑥log. Find the value of 𝑓(9).

Q7:

If 𝑔(𝑥) is the inverse of the function 𝑓(𝑥)=3, find 𝑔(𝑥).

  • A𝑔(𝑥)=𝑥2+3log
  • B𝑔(𝑥)=12𝑥−32log
  • C𝑔(𝑥)=12𝑥+32log
  • D𝑔(𝑥)=12𝑥−3log
  • E𝑔(𝑥)=2𝑥−3log

Q8:

If 𝑔(𝑥) is the inverse of the function 𝑓(𝑥)=2𝑒, find 𝑔(𝑥).

  • A𝑔(𝑥)=𝑥2−1ln
  • B𝑔(𝑥)=12(𝑥−1)ln
  • C𝑔(𝑥)=2𝑥−1ln
  • D𝑔(𝑥)=(2𝑥−1)ln
  • E𝑔(𝑥)=𝑥2−1ln

Q9:

Determine 𝑓(243), given that the graph of the function 𝑓(𝑥)=𝑥log passes through the point (81,4).

Q10:

The magnitude 𝑀(𝐼) of an earthquake on the Richter scale is given by 𝑀(𝐼)=𝐼𝐼log, where 𝐼 is the intensity of the earthquake and 𝐼 is a fixed reference intensity. What is the approximate intensity of an earthquake with a magnitude of 4.4 on the Richter scale?

  • A2,720,000 times the reference intensity
  • B25,000 times the reference intensity
  • C1,585 times the reference intensity
  • D2,512 times the reference intensity
  • E620,000 times the reference intensity

Q11:

Given that the graph of the function 𝑓(𝑥)=𝑥log passes through the point (1,024,5), find the value of 𝑎.

  • A20
  • B1
  • C√5
  • D9
  • E4

Q12:

The pH of a solution is given by the formula pHlog=−(𝑎)H+, where 𝑎H+ is the concentration of hydrogen ions. Determine the concentration of hydrogen ions in a solution whose pH is 8.4.

  • A10
  • B10
  • C10
  • D10

Q13:

The pH of a solution is given by the formula pHlog=−(𝑎)H+ where 𝑎H+ is the concentration of hydrogen ions. Determine the pH of a solution whose concentration of hydrogen ions is 10.

Q14:

Consider the function 𝑓(𝑥)=(3𝑥−1)log. If 𝑓(𝑎)=3, find the value of 𝑎.

Q15:

Consider the function 𝑓(𝑥)=𝑏, where 𝑏 is a positive real number not equal to 1. What is the domain of 𝑓(𝑥)?

  • AAll real numbers
  • B𝑥>𝑏
  • C0<𝑥<𝑏
  • D𝑥>0

Q16:

Consider the function 𝑓(𝑥)=(𝑥)+42log. Find the value of 𝑓(9).

  • A3
  • B52
  • C72
  • D4
  • E2

Q17:

If 𝑔(𝑥) is the inverse of the function 𝑓(𝑥)=52, find 𝑔(𝑥).

  • A𝑔(𝑥)=(2𝑥)−12log
  • B𝑔(𝑥)=(𝑥)−14log
  • C𝑔(𝑥)=(2𝑥)−14log
  • D𝑔(𝑥)=14(2𝑥)−1log
  • E𝑔(𝑥)=(2𝑥)+14log

Q18:

True or False: The domain of the function 𝑓(𝑥)=(2𝑥)−1log is 𝑥>0.

  • ATrue
  • BFalse

Q19:

In order to conduct a research on atherosclerosis, epithelial cells are harvested from discarded umbilical tissue and grown in the laboratory. A technician observes that a culture of nine thousand cells grows to three million cells in one week. Assuming that the cells follow the law of uninhibited growth, which states that 𝑁(𝑡)=𝑁𝑒, where 𝑁(0)=𝑁 is the initial number of organisms and 𝑘>0 is the constant of proportionality that satisfies the equation, find a formula for the number of cells, 𝑁 in thousands after 𝑡 days.

  • A𝑁(𝑡)=9𝑒ln
  • B𝑁(𝑡)=9𝑒ln
  • C𝑁(𝑡)=9𝑒ln
  • D𝑁(𝑡)=9𝑒ln
  • E𝑁(𝑡)=9𝑒ln

Q20:

Find the range of the function 𝑓(𝑥)=(2𝑥−4)log.

  • AAll real numbers
  • B𝑦<2
  • C𝑦>2
  • D𝑦>0
  • E𝑦>12

Q21:

If 𝑔(𝑥) is the inverse of the function 𝑓(𝑥)=2, find 𝑔(𝑥).

  • A𝑔(𝑥)=(𝑥)−2log
  • B𝑔(𝑥)=(𝑥)−5log
  • C𝑔(𝑥)=(𝑥)−5log
  • D𝑔(𝑥)=(𝑥)+5log
  • E𝑔(𝑥)=(𝑥)+2log

Q22:

Find the solution set of the equation log125=𝑥+1.

  • A{5}
  • B{6}
  • C{4}
  • D{3}
  • E{2}

Q23:

Find the domain of the following function: 𝑓(𝑥)=(𝑥+5)log.

  • A𝑥>5
  • B𝑥>−5
  • C𝑥<5
  • D𝑥<−5
  • EAll real numbers

Q24:

Iodine-131 is a commonly used radioactive isotope that helps detect how well the thyroid is functioning. Suppose that the decay of iodine-131 follows the radioactive decay model, where the amount of a radioactive element 𝐴 at time 𝑡 is given by the formula 𝐴(𝑡)=𝐴𝑒, 𝐴(0)=𝐴 is the initial amount of the element, and 𝑘>0 is the constant of proportionality that satisfies the equation, and that the half-life (the time it takes for half of the substance to decay) of iodine-131 is approximately 8 days. If 2 grams of iodine-131 is present initially, find a function that gives the amount of iodine-131, 𝐴, in grams ‏𝑡 days later.

  • A𝐴(𝑡)=2𝑒ln
  • B𝐴(𝑡)=2𝑒ln
  • C𝐴(𝑡)=2𝑒ln()
  • D𝐴(𝑡)=2𝑒ln
  • E𝐴(𝑡)=2𝑒()ln

Q25:

One hot water pump has a noise rating of 50 decibels. One dishwasher, however, has a noise rating of 62 decibels. If it is known that the noise rating 𝑑 of a sound is measured in a logarithmic scale in a unit called the decibel using the formula 𝑑=10𝑃𝑃log, where 𝑃 is the power or intensity of the sound and 𝑃 is the weakest sound that the human ear can hear, how many times is the dishwasher noise more intense than the hot water pump noise? Round your answer to the nearest hundredth.

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