log₅ 512 = log 512 log 5 = log 2⁹ log (10/2) = 9 log 2 log 10 - log 2 = (9 x 0.3010) 1 - 0.3010 = 2.709 0.699 = 2709 699 = 3.876

Answer : Option A

Explanation :

$MF#%\log_5{1024} = \dfrac{\log 1024}{\log 5} = \dfrac{\log\left(2^{10}\right)}{\log\left(\dfrac{10}{2}\right)}= \dfrac{10 \log(2)}{\log 10 - \log 2} $MF#%

$MF#%= \dfrac{10 \times 0.3010 }{1 - 0.3010} = \dfrac{3.01}{0.699} = \dfrac{3010}{699} = 4.31$MF#%

log ₂ x = 10         x = 2¹⁰.

log_x y = 100

y = x¹⁰⁰

y = (2¹⁰)¹⁰⁰     [put value of x]

y = 2¹⁰⁰⁰.

Let log₉ 27 = n
9ⁿ =27
3²ⁿ = 3³
2n=3
n=3/2
Again, let log₂₇9 = m
27^m = 9
3^3m =3²
3m=2
m=2/3
log₉27 - log₂₇9=(n-m)=(3/2-2/3)=5/6

log₂ 10 = 1 = 1 = 10000 = 1000 . log₁₀ 2 0.3010 3010 301

Answer : Option B

Explanation :

$MF#%\begin{align}&\dfrac{1}{3}\log_{10}125 - 2\log_{10}4 + \log_{10}32\\\\=
&\log_{10}\left(125^{1/3}\right) - \log_{10}\left(4^2\right) + \log_{10}32\\\\
&= \log_{10}5 - \log_{10} 16 + \log_{10}32\\\\
&= \log_{10}\left(\dfrac{5 \times 32 }{16}\right)\\\\
&= \log_{10}(10)\\\\
&= 1\end{align}$MF#%

Answer : Option C

Explanation :

log₂ 512 = log₂ (2⁹) = 9

log (2⁶⁴)
= 64 x log 2
= (64 x 0.30103)
= 19.26592

Its characteristic is 19.

Hence, then number of digits in 2⁶⁴ is 20.

log_x

9

= -
1
16
2

x^-1/2
=
9
16

1
=
9
x
16

x =
16
9

x =

16

2
9

x =
256
81

Let log₂ 16 = n.

Then, 2ⁿ = 16 = 2⁴         n = 4.

log₂ 16 = 4.