Worksheet: Logarithmic Functions

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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?

  • A 𝑎 = 1 2 , 𝑏 = − 3 2
  • B 𝑎 = − 3 2 , 𝑏 = 1 2
  • C 𝑎 = 1 2 , 𝑏 = 3 2
  • D 𝑎 = 1 , 𝑏 = − 3
  • E 𝑎 = 2 , 𝑏 = 3

Q2:

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

  • A 𝑥 = ( 𝑦 ) − 𝑏 𝑎 l o g
  • B 𝑥 = ( 𝑦 ) − 𝑏 l o g
  • C 𝑥 = ( 𝑦 − 2 ) − 𝑏 𝑎 l o g
  • D 𝑦 = ( 𝑥 ) − 𝑏 𝑎 l o g
  • E 𝑥 = ( 𝑦 ) + 𝑏 𝑎 l o g

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.

  • A 𝑔 ( 𝑦 ) = − 𝑦 − 1 𝑦 − 2 l o g
  • B 𝑔 ( 𝑥 ) = − 𝑥 − 1 𝑥 − 2 l o g
  • C 𝑔 ( 𝑥 ) = − 𝑥 − 1 𝑥 − 2 l o g
  • D 𝑔 ( 𝑥 ) = 𝑥 + 1 𝑥 − 2 l o g
  • E 𝑔 ( 𝑦 ) = − 𝑦 − 1 𝑦 − 2 l o g

Q4:

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
  • B 𝑥 > 𝑏
  • C 0 < 𝑥 < 𝑏
  • D 𝑥 > 0

Q5:

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

  • A 𝑘 = 1 , 𝐴 = 1 4
  • B 𝑘 = 1 , 𝐴 = 3 4
  • C 𝑘 = 3 , 𝐴 = 4
  • D 𝑘 = 3 4 , 𝐴 = 1
  • E 𝑘 = − 1 , 𝐴 = 1 1 2

Q6:

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

  • A 1 4
  • B 1 3
  • C − 3
  • D4
  • E3

Q7:

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

Q8:

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

  • A 𝑔 ( 𝑥 ) = 𝑥 2 + 3 l o g
  • B 𝑔 ( 𝑥 ) = 1 2 𝑥 − 3 2 l o g
  • C 𝑔 ( 𝑥 ) = 1 2 𝑥 + 3 2 l o g
  • D 𝑔 ( 𝑥 ) = 1 2 𝑥 − 3 l o g
  • E 𝑔 ( 𝑥 ) = 2 𝑥 − 3 l o g

Q9:

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

  • A 𝑔 ( 𝑥 ) = 𝑥 2 − 1 l n
  • B 𝑔 ( 𝑥 ) = 1 2 ( 𝑥 − 1 ) l n
  • C 𝑔 ( 𝑥 ) = 2 𝑥 − 1 l n
  • D 𝑔 ( 𝑥 ) = ( 2 𝑥 − 1 ) l n
  • E 𝑔 ( 𝑥 ) = 𝑥 2 − 1 l n

Q10:

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

Q11:

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

Q12:

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

Q13:

Determine the point at which the graph of the function 𝑓(𝑥)=(18−𝑥)log intersects the 𝑥-axis.

  • A ( 1 7 , 0 )
  • B ( 0 , 1 5 )
  • C ( 0 , 1 7 )
  • D ( 1 5 , 0 )
  • E ( 0 , 3 4 )

Q14:

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

  • A 1 0
  • B 1 0
  • C 1 0
  • D 1 0

Q15:

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

Q16:

The formula for calculating the magnitude of an earthquake is 𝑀=23𝑆𝑆log. One earthquake has magnitude of 3.9 on the MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second one. Round the answer to the nearest hundredth.

Q17:

Use technology to plot the graphs of 𝑓(𝑥)=𝑥 and 𝑔(𝑥)=𝑥ln. Find the coordinates where 𝑓(𝑥)=𝑔(𝑥) if the curves intersect, giving your answer to two decimal places.

  • AThey intersect at (0.45,0.20) and (0.14,0.02).
  • BThe curves do not intersect.
  • CThe curves are the same, so intersect everywhere.
  • DThey intersect at (0.37,0.00).
  • EThey intersect at (0.14,0.02).

Q18:

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

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