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_a(ab²) - log_a(ac) + log_a(abc⁴) - 3log_a(bc) = log_a(ab²) - log_a(ac) + log_a(abc⁴) - log_a(b³c³)

= \log_{a}(\frac{ab^{2}* abc^{4}}{ac * b^{3}c^{3}})


= log_aa = 1

Given,

log \frac{9}{8} - log \frac{27}{32} + log \frac{3}{4}

=  log [\frac{\frac{9}{8}}{\frac{27}{32}} \times \frac{3}{4}]

= log [\frac{9}{8} \times \frac{32}{27} \times \frac{3}{4}]

= log 1 = 0

Given Exp. = log\frac{75}{16} - 2 log\frac{5}{9} + log\frac{32}{243}
= log \frac{25 \times 3}{4 \times 4} - log \frac{25}{81} + log \frac{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

logX + logY = log(XY)
log \frac{a^{2}}{bc} + log \frac{b^{2}}{ac} + log \frac{c^{2}}{ab}
=log\frac{a^{2} ×b^{2} × c^{2}}{bc×ac×ab}
=log \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c^{2}}
=log1 = 0

log_a(x) +log_a(1+x) =  0

we know that log_a1 = 0

⇒ log_a(x) + log_a(1+x) =0    = log_a1

log_0.125(125) = 3\times \frac{1}{3}\times {log_{0.5}}5

log(5) = 0.699
log(0.5) = -0.3010 = log_0.1(2)
log_0.125(125) = log_0.5(5) = -2.3219

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

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

log₂(\frac{1}{64})= log₂(\frac{1}{64}) = log₂(64)^-1
= log₂(2⁶)^-1 = log₂2^-6
= (-6) log₂2 = −6×1 = -6

If \frac{log 125}{log 5} = x
then x = \frac{3log5}{log5}= 3