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 𝑎0.

  • A𝑥=(𝑦)𝑏𝑎log
  • B𝑥=(𝑦)𝑏log
  • C𝑥=(𝑦2)𝑏𝑎log
  • D𝑦=(𝑥)𝑏𝑎log
  • E𝑥=(𝑦)+𝑏𝑎log

Q3:

Rearrange the equation 𝑦=212+1 to find 𝑥 in terms of 𝑦. Hence determine the inverse 𝑔 to the function 𝑔(𝑥)=212+1.

  • A𝑔(𝑦)=𝑦1𝑦2log
  • B𝑔(𝑥)=𝑥1𝑥2log
  • C𝑔(𝑥)=𝑥1𝑥2log
  • D𝑔(𝑥)=𝑥+1𝑥2log
  • E𝑔(𝑦)=𝑦1𝑦2log

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
  • C3
  • 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)=12𝑥32log
  • C𝑔(𝑥)=12𝑥+32log
  • D𝑔(𝑥)=12𝑥3log
  • E𝑔(𝑥)=2𝑥3log

Q8:

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

  • A𝑔(𝑥)=𝑥21ln
  • B𝑔(𝑥)=12(𝑥1)ln
  • C𝑔(𝑥)=2𝑥1ln
  • D𝑔(𝑥)=(2𝑥1)ln
  • E𝑔(𝑥)=𝑥21ln

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
  • C5
  • 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)=(3𝑥1)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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.