44099.If $a^{x}=b^{y}$,then
$\log\dfrac{a}{b}=\dfrac{x}{y}$
$\dfrac{log a}{log b}=\dfrac{x}{y}$
$\dfrac{log a}{log b}=\dfrac{y}{x}$
None of these
Explanation:
$a^{x}$=$b^{y}$
=>$log a^{x}$=$log b^{y}$
=> x log a=y log b
=>$\dfrac{log a }{log b}$=$\dfrac{y}{x}.$
$a^{x}$=$b^{y}$
=>$log a^{x}$=$log b^{y}$
=> x log a=y log b
=>$\dfrac{log a }{log b}$=$\dfrac{y}{x}.$
44102.Which of the following statements is not correct?
$\log_{10}{10}$ = 1
log (2 + 3) = log (2 x 3)
$\log_{10}{1}$= 0
log (1 + 2 + 3) = log 1 + log 2 + log 3
Explanation:
(a)Since $\log_{a}{a}$=1,so $\log_{10}{10}$=1.
(b)log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3
=>log (2 + 3) $\neq $log (2 x 3)
(c) since,$\log_{a}{1}$=0,so $\log_{10}{1}$=0.
d) log (1 + 2 + 3) = log 6 = log (1 x 2 x 3) = log 1 + log 2 + log 3.
So, (b) is incorrect.
(a)Since $\log_{a}{a}$=1,so $\log_{10}{10}$=1.
(b)log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3
=>log (2 + 3) $\neq $log (2 x 3)
(c) since,$\log_{a}{1}$=0,so $\log_{10}{1}$=0.
d) log (1 + 2 + 3) = log 6 = log (1 x 2 x 3) = log 1 + log 2 + log 3.
So, (b) is incorrect.
44103.If $\log_{10}{2}=0.3010,then \log_{2}{10}$ is equal to :
$\dfrac{699}{301}$
$\dfrac{1000}{301}$
0.3010
0.6990
Explanation:
$\log_{2}{10}$=$\dfrac{1}{\log_{10}{2}}$
=>$\dfrac{1}{0.3010}$
=>$\dfrac{10000}{3010}$
=>$\dfrac{1000}{301}$
$\log_{2}{10}$=$\dfrac{1}{\log_{10}{2}}$
=>$\dfrac{1}{0.3010}$
=>$\dfrac{10000}{3010}$
=>$\dfrac{1000}{301}$
44105.If log 2 = 0.30103, the number of digits in $2^{64} $is:
18
19
20
21
Explanation:
$\log_(2^{64})$=$64 \times log 2$
=$(64 \times 0.30103)$
=19.26592
Its characteristic is 19.
Hence then number of digits in $2^{64}$ is 20.
$\log_(2^{64})$=$64 \times log 2$
=$(64 \times 0.30103)$
=19.26592
Its characteristic is 19.
Hence then number of digits in $2^{64}$ is 20.
44107.If $\log_{10}{2}=0.3010,then \log_{10}{80}$ is equal to :
1.6020
1.9030
3.9030
None of these
Explanation:
$\log_{10}{80}$=$\log_{10}{8 \times 10}$
=$\log_{10}{8}+\log_{10}{10}$
=$\log_{10}{2^{3}}+1$
=$3 \log_{10}{2}+1$
=$(3 \times 0.3010)+1$
=1.9030
$\log_{10}{80}$=$\log_{10}{8 \times 10}$
=$\log_{10}{8}+\log_{10}{10}$
=$\log_{10}{2^{3}}+1$
=$3 \log_{10}{2}+1$
=$(3 \times 0.3010)+1$
=1.9030
44109.If $\log_x(\frac{9}{16})=-\frac{1}{2}$,then x is equal to :
$-\dfrac{3}{4}$
$\dfrac{3}{4}$
$\dfrac{81}{256}$
$\dfrac{256}{81}$
Explanation:
$\log_(\dfrac{9}{16})$=$-\dfrac{1}{2}$
=>$x^{\dfrac{-1}{2}}$=$\dfrac{9}{16}$
=>$\dfrac{1}{\sqrt{x}}$=$\dfrac{9}{16}$
=>$\dfrac{1}{\sqrt{x}}$=$\dfrac{16}{9}$
=>$\sqrt{x}$=$\dfrac{16}{9}$
=>x=$(\dfrac{16}{9})^{2}$
=>x=$\dfrac{256}{81}$
$\log_(\dfrac{9}{16})$=$-\dfrac{1}{2}$
=>$x^{\dfrac{-1}{2}}$=$\dfrac{9}{16}$
=>$\dfrac{1}{\sqrt{x}}$=$\dfrac{9}{16}$
=>$\dfrac{1}{\sqrt{x}}$=$\dfrac{16}{9}$
=>$\sqrt{x}$=$\dfrac{16}{9}$
=>x=$(\dfrac{16}{9})^{2}$
=>x=$\dfrac{256}{81}$
44110.$\dfrac{\log\sqrt{8}}{log 8}$ is equal to
$\dfrac{1}{\sqrt{8}}$
$\dfrac{1}{4}$
$\dfrac{1}{2}$
$\dfrac{1}{8}$
Explanation:
$\dfrac{\log\sqrt{8}}{log 8}$
=>$\dfrac{log8{\dfrac{1}{2}}}{log 8}$
=$\dfrac{\dfrac{1}{2}log8}{log 8}$
=$\dfrac{1}{2}$
$\dfrac{\log\sqrt{8}}{log 8}$
=>$\dfrac{log8{\dfrac{1}{2}}}{log 8}$
=$\dfrac{\dfrac{1}{2}log8}{log 8}$
=$\dfrac{1}{2}$
44111.If $\log{x}{y}=100\: and \: \log{2}{x}=10$,then the value of y is
$2^{10}$
$2^{100}$
$2^{1000}$
$2^{10000}$
Explanation:
$\log{2}{x}$=10=>x=$2^{10}$
therefore,$\log{x}{y}$=100
=>y=$x^{100}$
=>y=$(2^{10})^{100}$
=>y=$2^{1000}$
$\log{2}{x}$=10=>x=$2^{10}$
therefore,$\log{x}{y}$=100
=>y=$x^{100}$
=>y=$(2^{10})^{100}$
=>y=$2^{1000}$
44112.If $log 2 = 0.3010 \:and \:log 3 = 0.4771,\: the\: value \:of \:\log_{5}{512} $is:
2.870
2.967
3.876
3.912
Explanation:
$\log_{5}{512}$=$\dfrac{log 512}{log 5}$
=$\dfrac{\log2^9}{\log(\dfrac{10}{2})}$
=$\dfrac{9 log 2}{log 10 -log 2}$
=$\dfrac{(9\times 0.3010}{1-0.3010}$
=$\dfrac{2.709}{0.699}$
=$\dfrac{2709}{699}$
=3.876
$\log_{5}{512}$=$\dfrac{log 512}{log 5}$
=$\dfrac{\log2^9}{\log(\dfrac{10}{2})}$
=$\dfrac{9 log 2}{log 10 -log 2}$
=$\dfrac{(9\times 0.3010}{1-0.3010}$
=$\dfrac{2.709}{0.699}$
=$\dfrac{2709}{699}$
=3.876
44113.If log 27 = 1.431, then the value of log 9 is:
0.934
0.945
0.954
0.958
Explanation:
log 27 = 1.431
=$log 3^{3}$=1.431
=3 log 3 = 1.431
= log 3 = 0.477
=log 9=$log(3^{2})$=2log3=$(2 \times 0.477)$=0.954.
log 27 = 1.431
=$log 3^{3}$=1.431
=3 log 3 = 1.431
= log 3 = 0.477
=log 9=$log(3^{2})$=2log3=$(2 \times 0.477)$=0.954.