Worksheet: Logarithmic Functions

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

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