# LOGARITHMS

Ans .

3,-3,-1

1. Explanation :

(1) let LOG(3, 27)=$$3^3$$ or n=3.

ie, $$log_3$$(27)= 3.

(2) Let $$log_7$$(1\343) = n.

Then ,7n =1/343=1/73

n = -3.ie,log7(1\343)= -3.

(3) let $$log_{100}$$(0.01) = n.

Then,. (100) = 0.01 = 1 /100=100 -1 0r n=-1

Ans .

0,0,16

1. Explanation :

solution: i) we know that $$log_a$$ 1=0 ,so $$log_7$$ 1=0 . ii) we know that $$log_a$$a=1,so $$\log_{34} 34$$ =0. iii) We know that $$a^{\log_{6} x}$$ =x.

now $$36^{\log_{6} 4}$$=$$6^{2^{\log_{6} 4}}$$ =$$6^{\log_{6} 16}$$=16.

Ans .

32

1. Explanation :

$$\log_{\sqrt{8}} x$$=10/3

x=$$\sqrt{8}^{10/3}$$ =$$2^{\frac{2}{3} ^{\frac{10}{3}}}$$=$$2^{\frac{2}{3} * \frac{10}{3}}$$=25=32.

Ans .

2/3,5/6

1. Explanation :

$$\log_{5} 3$$* $$\log_{27} 25$$=log 3/log 5*log 25/log 27 =(log 3 /log 5) * log$$5^2$$ *log$$3^3$$

=(log 3/log 5)*(2log 5 / 3(log 3)

=2/3

(ii)Let $$\log_{9} 27$$=n

Then,$$9 ^n$$ =27 $$3^{2n}$$ =$$3^3$$ 2n=3 n=3/2

Again, let $$\log_{27} 9$$=m

Then,$$27^m$$ =9 $$3^{3m}$$ =$$3 ^2$$ 3m=2 m=2/3

$$\log_{9} 27$$- $$\log_{27} 9$$=(n-m)=(3/2-2/3)=5/6

Ans .

log2

1. Explanation :

log 75/16-2 log 5/9+log 32/243

= log 75/16-log(5/9)2+log32/243

= log 75/16-log25/81+log 32/243

= log(75/16*32/243*81/25)=log 2

Ans .

7/2

1. Explanation :

$$\log_{10} 3$$+$$\log_{10} (4x+1)$$=$$\log_{10} (x+1)$$+1

$$\log_{10} 3$$+$$\log_{10} (4x+1)$$=$$\log_{10} (x+1)$$+$$\log_{10} (x+1)$$+$$\log_{10} 10$$

$$\log_{10} (3(4x+1))$$=$$\log_{10} (10(x+1))$$ =3(4x+1)=10(x+1)=12x+3 =10x+10 =2x=7=x=7/2

Ans .

2

1. Explanation :

$$\log_{xyz} (xy)$$ + $$\log_{xyz} (yz)$$ + $$\log_{xyz} (zx)$$

=$$\log_{xyz} (xy*yz*zx)$$=$$\log_{xyz} {(xyz)^2}$$ 2$$\log_{xyz} (xyz)$$ =2*1=2

Ans .

1.69897

1. Explanation :

log 50=log (100/2)=log 100-log 2=2-0.30103=1.69897.

Ans .

1.39,0.65

1. Explanation :

i) log 25=log(100/4)=log 100-log 4=2-2log 2=(2-2*.3010)=1.398.

ii) log 4.5=log(9/2)=log 9-log 2=2log 3-log 2=(2*0.4771-.3010)=0.6532

Ans .

17

1. Explanation :

log $$2^{56}$$ =56 log2=(56*0.30103)=16.85768. Its characteristics is 16. Hence,the number of digits in $$2^{56}$$ is 17