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# The Importance of Being Earnest

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Question 1. Find the inverse of the function h(x) = x2+3, where x>=0.

Answer: h(x) is a one to one function, representing it as y :

y = 2x+3

implies x = (y-3)/2

Hence h-1 (x) = (x-3)/2 with domain restriction of x>= 3 to keep h-1 (x) >=0.

Question 2. Evaluate :

(a) Log 4 2

Answer: by definition of logarithm, if log 4 2 be x :

4 x = 2

Implies 2 2x = 2, since the bases are equal :

2x=1,

x = ½, hence Log 4 2 =1/2

(b) Log913

Answer : using Logbx = logax/logab, we can write

Log913 = In 13/ In 9

= 2.5649/2.1972

= 1.1673

(c) In e3

Answer : using Log (xy) =Log x+Log y, we can write

In e3 = In e + In 3

= 1+1.0986

= 2.0986

(d) Log 1/5 25

Answer: by definition of logarithm, we can write

(1/5)x = 25, where x=Log 1/5 25

Implies (1/5)x =(1/5)-2  , as (1/5)-2 =25

Implies x= -2

Question 3. Solve 23x-1 =16.

Adding 1 to both sides :

23x=16+1

23x=17

Dividing both sides by 23:

x= 17/23

Question 4. Solve for x, log 5 (3x-4) = log 5(2x+7).

Answer: since logarithms are unique and one is to one function, we have

Log 5(3x-4) = Log 5(2x+7), implies

3x-4 = 2x+7, as logarithmic base is same

3x-2x=7+4, rearranging for x

x= 11

September 25, 2023
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