Logarithm - Study Mode

[#41] $$frac{1}{{{{log }_a}b}} imes frac{1}{{{{log }_b}c}} imes frac{1}{{{{log }_c}a}}$$ xa0 xa0 is equal to -
Correct Answer

(D) 1

Explanation

Solution: $$eqalign{
& { ext{Given}},,,{ ext{Expression}} cr
& left( {frac{{log a}}{{log b}} imes frac{{log b}}{{log c}} imes frac{{log c}}{{log a}}}
ight) cr
& = 1 cr} $$

[#42] $${frac{1}{{left( {{{log }_a}bc}
ight) + 1}} + }$$ xa0 $${frac{1}{{left( {{{log }_b}ca}
ight) + 1}} + }$$ xa0 $${frac{1}{{left( {{{log }_c}ab}
ight) + 1}}}$$ xa0 is equal to -
Correct Answer

(A) 1

Explanation

Solution: $$eqalign{
& { ext{Given}},,,{ ext{Expression}} cr
& = frac{1}{{{{log }_a}bc + {{log }_a}a}} + frac{1}{{{{log }_b}ca + {{log }_b}b}} + frac{1}{{{{log }_c}ab + {{log }_c}c}} cr
& = frac{1}{{{{log }_a}left( {abc}
ight)}} + frac{1}{{{{log }_b}left( {abc}
ight)}} + frac{1}{{{{log }_c}left( {abc}
ight)}} cr
& = {log _{abc}}a + {log _{abc}}b + {log _{abc}}c cr
& = {log _{abc}}left( {abc}
ight) cr
& = 1 cr} $$

[#43] If $${log _{10}}7 = a,$$ xa0 then $${log _{10}}left( {frac{1}{{70}}}
ight)$$ xa0 is equal to -
Correct Answer

(A) -(1 + a)

Explanation

Solution: $$eqalign{
& {log _{10}}left( {frac{1}{{70}}}
ight) cr
& = {log _{10}}1 - {log _{10}}70 cr
& = - {log _{10}}left( {7 imes 10}
ight) cr
& = - left( {{{log }_{10}}7 + {{log }_{10}}10}
ight) cr
& = - left( {a + 1}
ight) cr} $$

[#44] If $$log x - 5log 3 = - 2,$$ xa0 xa0 then x equals -
Correct Answer

(D) 2.43

Explanation

Solution: $$eqalign{
& log x - 5log 3 = - 2 cr
& Rightarrow log x - log {3^5} = - 2 cr
& Rightarrow log left( {frac{x}{{{3^5}}}}
ight) = - 2 cr
& Rightarrow frac{x}{{243}} = {10^{ - 2}} = frac{1}{{100}} cr
& Rightarrow x = frac{{243}}{{100}} = 2.43 cr} $$

[#45] If $$a = {b^2} = {c^3} = {d^4},$$ xa0xa0 then the value of $${log _a}left( {abcd}
ight)$$ xa0 would be -
Correct Answer

(C) $${ ext{1 + }}frac{1}{2} + frac{1}{3} + frac{1}{4}$$

Explanation

Solution: $$eqalign{
& a = {b^2} = {c^3} = {d^4} cr
& Rightarrow b = {a^{frac{1}{2}}},,,,,c = {a^{frac{1}{3}}},,,,,d = {a^{frac{1}{4}}} cr
& herefore {log _a}left( {abcd}
ight) cr
& = {log _a}left( {a imes {a^{frac{1}{2}}} imes {a^{frac{1}{3}}} imes {a^{frac{1}{4}}}}
ight) cr
& = {log _a}{a^{left( {1 + frac{1}{2} + frac{1}{3} + frac{1}{4}}
ight)}} cr
& = left( {1 + frac{1}{2} + frac{1}{3} + frac{1}{4}}
ight){log _a}a cr
& = 1 + frac{1}{2} + frac{1}{3} + frac{1}{4} cr} $$