How many digits are there in (54)10 ?
(Given that
log102 = 0.301 and
log103 = 0-477)
Answers
Step-by-step explanation:
\red{ \huge{ \underline{ \overline{ \mid{ \bold{ \purple{ \: \: SOLUTION \: \red{ \mid}}}}}}}}
\underline{ \bold{ \: \: GIVEN \: : }} \to \: \: \: \: \: \: \: \bold{ log_{10}3 =0 .477} \\ \\ \\ \underline{ \bold{ \: \: TO\: \: FIND\: \: }} \to \: \: \bold{No. \: \: of \: \: digit \: \: in \: \: 3 {}^{40} =? }
\bold{Let,} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{x = 3 {}^{40} } \\ \\ \underline{\bold { \: \: Taking \: \: logarithm \: \: in \: \: both \: \: sides \: }} \\ \\ \: \: \: \: \: \: \: \: \: \: \bold{ log_{10}x = log_{10}(3 {}^{40} ) } \\ \\ \bold{ \Longrightarrow \: \: log_{10}x = 40 \: \: log_{10}3 } \\ \\ \bold{ \Longrightarrow \: \: log_{10}x = 40 \times 0.477} \\ \\ \bold{ \Longrightarrow \: \: log_{10}x = 19.08}
\bold{ \: \: Since ,\: \: the \: \: characteristic \: \: of \: \: } \\ \\ \: \: \: \: \: \: \bold{ log_{10}x = log_{10}3 {}^{40} = 19} \\ \\ \bold{So, \: \: the \: \: number \: \: of \: \: digit \: \: in} \\ \\ \bold{x = 3 {}^{40} = 19 + 1 = \boxed{ \bold{ \: \: 20 \: \: }}}
\bold{Hence,} \\ \\ \bold{If \: \: log_{10}3 = 0.477 \: \: , \: then \: \: the \: \: no. \: \: of \: \: } \\ \bold{digit \: \: in \: \: 3 {}^{40} = \underline{ \red{ \: \: 20 \: \: }}}
Answer is
Log10(54)^10 = log10(2^10×3^30)
=Log10(2)^10+log(3)^30
=10log10(2)+30log10(3)
Put the value of log10(2) and log10(3) in this equation
Now log10(54)^10 = 10(0.301)+30(0.477)
=3.01+14.41
=17.42
By the rule of log10(n) ..add 1 in the given answer
Hence 17.42+1=18
Answer is 18