Logarithm - Study Mode

[#11] If $${log _{10000}}x = - frac{1}{4}{ ext{,}}$$ xa0xa0 then the value of x is = ?
Correct Answer

(A) $$frac{1}{{10}}$$

Explanation

Solution: $$eqalign{
& { ext{let }}{log _{10000}}x = - frac{1}{4} cr
& Rightarrow x = {left( {10000}
ight)^{ - frac{1}{4}}} cr
& = {left( {{{10}^4}}
ight)^{ - frac{1}{4}}} cr
& = {10^{ - 1}} cr
& = frac{1}{{10}} cr} $$

[#12] $$frac{{log sqrt 8 }}{{log 8}},,{ ext{is equal to = ?}}$$
Correct Answer

(C) $$frac{1}{2}$$

Explanation

Solution: $$eqalign{
& frac{{log sqrt 8 }}{{log 8}} cr
& = frac{{log {{left( 8
ight)}^{frac{1}{2}}}}}{{log 8}} cr
& = frac{{frac{1}{2}log 8}}{{log 8}} cr
& = frac{1}{2} cr} $$

[#13] The value of $$frac{{6{ ext{ }}{{log }_{10}}1000}}{{3{ ext{ }}{{log }_{10}}100}}$$ xa0 is equal to -
Correct Answer

(D) 3

Explanation

Solution: $$eqalign{
& frac{{6{ ext{ }}{{log }_{10}}1000}}{{3{ ext{ }}{{log }_{10}}100}} cr
& = frac{{6{ ext{ }}{{log }_{10}}{{10}^3}}}{{3{ ext{ }}{{log }_{10}}{{10}^2}}} cr
& = frac{{6 imes 3{ ext{ }}{{log }_{10}}10}}{{3 imes 2{ ext{ }}{{log }_{10}}10}} cr
& = frac{{18}}{6} cr
& = 3 cr} $$

[#14] If $${log _2}left[ {{{log }_3}left( {{{log }_2}x}
ight)}
ight] = 1,$$ xa0 xa0 then x is equal to = ?
Correct Answer

(D) 512

Explanation

Solution: $$eqalign{
& {{log }_2}left[ {{ ext{ }}{{log }_3}left( {{ ext{ }}{{log }_2}x}
ight)}
ight] = 1 cr
& Rightarrow {log _3}left( {{{log }_2}x}
ight) = {2^1} = 2 cr
& Rightarrow {log _2}x = {3^2} = 9 cr
& Rightarrow x = {2^9} = 512 cr} $$

[#15] What is the value of the following expression? $$log left( {frac{9}{{14}}}
ight) - $$ xa0 $$log left( {frac{{15}}{{16}}}
ight) + $$ xa0 $$log left( {frac{{35}}{{24}}}
ight)$$
Correct Answer

(A) 0

Explanation

Solution: $$eqalign{
& log left( {frac{9}{{14}}}
ight) - log left( {frac{{15}}{{16}}}
ight) + log left( {frac{{35}}{{24}}}
ight) cr
& = log left( {frac{9}{{14}} div frac{{15}}{{16}} imes frac{{35}}{{24}}}
ight) cr
& = log left( {frac{9}{{14}} imes frac{{16}}{{15}} imes frac{{35}}{{24}}}
ight) cr
& = log 1 cr
& = 0 cr} $$