Cognizant Logarithm Quiz-1

log ₂ x = 10 ⇒ x = 2¹⁰.
∴ log _x y = 100
⇒ y = x¹⁰⁰
⇒ y = (2¹⁰)¹⁰⁰ [put value of x]
⇒ y = 2¹⁰⁰⁰.

log(ab²) - log(ac) + log(abc⁴) - 3log(bc)
= log(ab²) - log(ac) + log(abc⁴) - log(b³c³)
= log_a(ab²/ac)*(abc⁴/b³c³)
= log_aa = 1

Given Exp. = log [{(9/8) / (27/32)} x 3/4)]
= log [(9/8) x (3/4) x (32/27)]
= log 1
= 0

Given Exp. = log75/16 - 2 log5/9 + log32/243
= log [(25 x 3) / (4 x 4)] - log (25/81) + log [(16 x 2) / (81 x 3)]
= log(25 x 3) - log ( 4 x 4 ) - log(25) + log81 + log(16 x 2) -log (81 x 3)
= log 25 + log 3 - log 16 - log 25 + log 81 + log 16 + log 2 - log 81 - log 3
= log 2

log(a.a/bc)+log(b.b/ac)+log(c.c/ab)

log(a.a.b.b.c.c)/(b.c.a.c.a.b) = log1 = 0

loga(x) +loga(1+x) = 0

we know that log a1 =0

=>log a (x) + log (1+x) =0
=>log a (x) + log (1+x) = log a 1

Exp. = log_0.125125 = log₂^-3 5³ = - 3/3log₂5 = -log₂5
∵ 10^0.3010 = 2
⇒ log₁₀2 = 0.3010
∵ log₁₀5 = log₁₀10/2 = log₁₀10-log₁₀2
= 1-0.3010 = 0.6990
∴ -log₂5 = - log₁₀⁵ / log₁₀²
= - 0.6990 / 0.3010 = - 699/301

Explanation log10(10)=1

log10(10)+log10(100)+log10(1000)+log10(10000)+log10(100000)

[from log(a^2)=2loga]

=log10(10)+2log10(10)+3log10(10)+4log10(10)+5log10(10)

=1+2+3+4+5=15

Log₂ (1/64)= -log₂ (64) = -log₂ (2^6) = -6log₂(2)= -6

If log 125 / log 5 = x
then x = 3log5 / log5= 3