Having given $\log 3=0.4771213$, find the number of digits in
(1) ${{3}^{43}}$, (2) ${{3}^{27}}$, and (3) ${{3}^{62}}$
and the position of the first significant figure in
(4) ${{3}^{-13}}$, (5) ${{3}^{-43}}$, and (6) ${{3}^{-65}}$ \[\]

Hint: We recall the definition of characteristics of mantissa of logarithm. We find the number of digits present in the expansion of ${{a}^{n}}$ as characteristics plus 1 in the decimal expression of $n\log a$ and the position of the first significant figure in the expansion of ${{a}^{-n}}, a>1$ as the characteristics appearing in the bar notation of $-n\log a$.

Complete step-by-step solution
We know that in base-10 logarithms when we express the logarithm values in decimals the fractional part is called mantissa and the integral part is called the characteristics. In $\log 120\simeq 2.07918$ the mantissa is the fractional part $0.07918$ and 2 is the characteristics.
We know that the number of digits present in ${{a}^{n}}$ is equal to characteristics plus 1 in the decimal expression of $n\log a$. \[\]
We know that significant digits of a number written in positional notation are digits that carry meaningful contributions. We know that the position of the first significant figure of ${{a}^{-n}}$ is the integer appearing in the bar notation of $-n\log a$. \[\]
(i) Let us take logarithm of given number ${{3}^{43}}$ and have $\log {{3}^{43}}=43\times \log 3=43\times 0.4771213=20.5162159$. So here the characteristic is 20 and the number of digits in ${{3}^{43}}$ is $20+1=21$\[\]
(ii) Let us take logarithm of given number ${{3}^{27}}$ and have $\log {{3}^{27}}=27\times \log 3=27\times 0.4771213=12.8822751$. So here the characteristic is 27 and the number of digits in ${{3}^{27}}$ is $12+1=13$.\[\]
(iii) Let us take logarithm of given number ${{3}^{62}}$ and have $\log {{3}^{62}}=62\times \log 3=62\times 0.4771213=29.5815206$. So here the characteristic is 29 and the number of digits in ${{3}^{62}}$ is $29+1=30$.\[\]
 (iv) Let us take logarithm of given number ${{3}^{-13}}$ and have $\log {{3}^{-13}}=-13\times \log 3=-13\times 0.4771213=-6.2025769$. We convert the decimal into bar notation as $\log {{3}^{-13}}=-6.2025769=-7+1-0.2025769=\overline{7}.7974231$. So the first significant figure will occur at in the seventh places in the decimals.\[\]
(v) Let us take logarithm of given number ${{3}^{-43}}$ and have $\log {{3}^{-43}}=-43\times \log 3=-43\times 0.4771213=-20.5162159$. We convert the decimal into bar notation as $\log {{3}^{-43}}=-20.5162159=-21+1-0.5162159=\overline{21}.4837841$. So the first significant figure will occur in the twenty-first places in the decimals. \[\]
(vi) Let us take logarithm of given number ${{3}^{-65}}$ and have $\log {{3}^{-65}}=-65\times \log 3=-65\times 0.4771213=-31.01228845$. We convert the decimal into bar notation as $\log {{3}^{-65}}=-31.01228845=-32+1-0.01228845=\overline{32}.9871155$. So the first significant figure will occur at in the seventh places in the decimals.\[\]

Note: We note that we have frequently used logarithmic identity $\log {{x}^{n}}=n\log x,x>0,n\ne 0$. We do not count leading zeros or trailing zeros unless they are in between two meaningful digits or over-lined in significant figures. If we know the greatest integer function $\left[ x \right]$which returns greatest integer less than equal to $x$ the we can find the number of digits in ${{a}^{n}}$ as $D=\left[ {{\log }_{10}}{{a}^{n}} \right]+1$and position of first significant figure in ${{a}^{-n}}$ as $D=\left[ {{\log }_{10}}{{a}^{-n}} \right]$.