# Exam answers to indices and logarithm question(MTH111)

1. XLog3base3 = Log3base7
Recall, Log3 base 3 = 1 (Log of the same base = 1)
X ร 1 = Log3base7
X = Log3base7 ( Find Log 7 base 3 )
X = 1.78
(ii) 2^x + 2^xโยน = 2
Solution
2^x + 1/2^x= 2 [ any number raise to power minus sign(-) is inverse of that number]2^x + 1/2^X = 2 ( find l.c.m)
(4xยฒ + 1) รท2x =2 [ cross multiply ]4ยฒ + 1 =2 ร 2x
4xยฒ + 1 = 4x
4xยฒ โ 4x + 1 =0
Using factorization method,
(4xยฒ โ 2x) โ( 2x + 1) =0 [ hint: whatโs common in the both side]2x (2x โ 1) -1(2x-1) = 0
(2x โ 1)(2x-1) = 0
(2x โ 1) appears twice
2x โ 1 = 0
2x = 1 [ divide both side by 2]X = ยฝ(twice)

To prove, put X as ยฝ into the real equation
2^X + 1/2^x = 2
2^ยฝ+ 1/2^ยฝ= 2
1.414 + 0.707 = 2 ( approximately)

## b. simplify Log1/5125 (ii) Prove that Logba ร Logcb ร Logac = 1solution

C. If Log2 base5= 0.431 and Log3base 5 =0.682, evaluate Log54/9
Solution
evaluate Log54/9 = Log54 รท Log59
Log522 รท Log532 = 2Log52 รท 2Log53
(2 ร 0.431) รท ( 2 ร 0.682)
0.862/1.364 = 0.632( approximately)

### Log1/5125 = Log5-1 125

Let the both be in base 5
Log5125 รท Log55-1
Log553 รท Log55-1
3Log55 รท -1Log55
3รท-1 = -3

### (ii) Prove that Logba ร Logcb ร Logac = 1

Weโll also apply the same thing to this, only that their bases will be different
Logba ร Logcb ร Logac
(Logaa รท Logbb) ร (Logbb รท Logcc) ร( Logcc รท Logaa)
As we know Log of the same base = 1
1/1 ร 1/1 ร 1/1 = 1 ร 1 ร 1
=1

Question on surd

2-1/(โ2-1)
Using rationalization method,

2-1/(โ2-1) ร (โ2+1)/(โ2+1)
2- (โ2+1)/ (2+โ2-โ2-1)
2-(โ2+1)/1
2-โ2-1
1-โ2

YOU CAN DO IT

### Oluwamuyide Peter

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