Logarithms : Exercise (3.1) Solutions


  • တိုင်းတာမှုများပါသော တွက်ချက်မှုများ ဆောင်ရွက်ရာတွင် ရလဒ်ကို တိကျမှုအနည်းဆုံး ဖြစ်သော အတိုင်းအတာပမာဏအတိုင်း ဖေါ်ပြပေးသင့်ပါသည်။

  • ပေါင်းခြင်း နှင့် နုတ်ခြင်းတွင် ရလဒ်ကို တိကျမှုအနည်းဆုံး ဖြစ်သော အတိုင်းအတာ၏ ဒသမ အရေအတွက်အတိုင်း ဖေါ်ပြပေးသင့်ပါသည်။

  • မြှောက်ခြင်းနှင့် စားခြင်းတွင် ရလဒ်ကို တိကျမှုအနည်းဆုံး ဖြစ်သော အတိုင်းအတာ၏ အရာရောက်ဂဏန်း အရေအတွက်အတိုင်း ဖေါ်ပြပေးသင့်ပါသည်။

  • The Rules for Significant Figures


    $\begin{array}{|l|l|l|l|} \hline \text { Example } & \begin{array}{c} \text { Scientific } \\ \text { Notation } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Significant } \\ \text { Figures } \end{array} & {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { Remark }} \\ \hline 0.00682 & 6.82 \times 10^{-3} & \ \ \ \ \ \ \ \ 3 & \text { Leading zeros are not significant. } \\ \hline 100.07 & 1.0007 \times 10^{2} & \ \ \ \ \ \ \ \ 5 & \text { Embedded zeros are always significant. } \\ \hline 300 & 3 \times 10^{2} & \ \ \ \ \ \ \ \ 1 & \begin{array}{l} \text { Trailing zeros are significant only if the } \\ \text { decimal point is specified. If the decimal } \\ \text { point is not specified, trailing zeros are } \\ \text { generally not significant. } \end{array} \\ \hline 300 . & 3.00 \times 10^{2} & \ \ \ \ \ \ \ \ 3 & \text { All digits are significant. } \\ \hline 300.0 & 3.000 \times 10^{3} & \ \ \ \ \ \ \ \ 4 & \text { All digits are significant. } \\ \hline 0.0080500 & \begin{array}{l} 8.0500 \times10^{-3} \\ \end{array} & \ \ \ \ \ \ \ \ 5 & \begin{array}{l} \text { Leading zeros are not significant whereas } \\ \text { embedded zeros and trailing zeros after } \\ \text { the decimal point are significant. } \end{array} \\ \hline \end{array}$

    1.           How many significant figures are there in each of the following numbers?

                  $\begin{array}{lll} \text{(a)}\ \ 2.175 & \text{(b)}\ \ 0.2175& \text{(c)}\ \ 0.0075\\\\ \text{(d)}\ \ 89400 & \text{(e)}\ \ 0.00046 & \text{(f)}\ \ 12.0500 & \end{array}$

    Show/Hide Solution

    $\begin{array}{lll} \text{(a)}\ \ 2.175 & \Rightarrow & 4\ \text{significant figures}\\\\ \text{(b)}\ \ 0.2175 & \Rightarrow & 4\ \text{significant figures}\\\\ \text{(c)}\ \ 0.0075 & \Rightarrow & 2\ \text{significant figures}\\\\ \text{(d)}\ \ 89400 & \Rightarrow & 3\ \text{significant figures}\\\\ \text{(e)}\ \ 0.00046 & \Rightarrow & 2\ \text{significant figures}\\\\ \text{(f)}\ \ 12.0500 & \Rightarrow & 6\ \text{significant figures}\\\\ \end{array}$

    2.           Write in scientific notation.

                  $\begin{array}{lll} \text{(a)}\ \ 24.86 & \text{(b)}\ \ 2.486& \text{(c)}\ \ 0.2486\\\\ \text{(d)}\ \ 0.002486 & \text{(e)}\ \ 0.073 & \text{(f)}\ \ 0.0086\\\\ \text{(g)}\ \ 0.934 & \text{(h)}\ \ 7 & \text{(i)}\ \ 6.843250\\\\ \text{(j)}\ \ 0.00056857 & \text{(k)}\ \ 62500 & \text{(l)}\ \ 3001 \end{array}$

    Show/Hide Solution

    $\begin{array}{lll} \text{(a)}\ \ 24.86 & = & 2.486\times 10^1 \\\\ \text{(b)}\ \ 2.486 & = & 2.486\times 10^0 \\\\ \text{(c)}\ \ 0.2486 & = & 2.486\times 10^{-1} \\\\ \text{(d)}\ \ 0.002486 & = & 2.486\times 10^{-3} \\\\ \text{(e)}\ \ 0.073 & = & 7.3\times 10^{-2} \\\\ \text{(f)}\ \ 0.0086 & = & 8.6\times 10^{-3} \\\\ \text{(g)}\ \ 0.934 & = & 9.34\times 10^{-1} \\\\ \text{(h)}\ \ 7 & = & 7\times 10^0 \\\\ \text{(i)}\ \ 6.843250 & = & 6.843250\times 10^0 \\\\ \text{(j)}\ \ 0.00056857& = & 5.6857\times 10^{-4} \\\\ \text{(k)}\ \ 62500 & = & 6.25\times 10^{4} \\\\ \text{(l)}\ \ 3001 & = & 3.001\times 10^3 \end{array}$

    3.           Write each number in ordinary decimal form.

                  $\begin{array}{l} \text{(a)}\ \ 7.84 \times 10^{4}\\\\ \text{(b)}\ \ 7.89 \times 10^{-4}\\\\ \text{(c)}\ \ 2.25 \times 10^{5}\\\\ \text{(d)}\ \ 4.01 \times 10^{-3} \end{array}$

    Show/Hide Solution

    $\begin{array}{lll} \text{(a)}\ \ 7.84 \times 10^{4} &=& 78400\\\\ \text{(b)}\ \ 7.89 \times 10^{-4} &=& 0.000789\\\\ \text{(c)}\ \ 2.25 \times 10^{5} &=& 225000\\\\ \text{(d)}\ \ 4.01 \times 10^{-3} &=& 0.00401 \end{array}$

    4.           Simplify and give the answers in scientific notation.

                  $\begin{array}{l} \text{(a)}\ \ 2.3 \times 10^{2}+1.7 \times 10^{2}\\\\ \text{(b)}\ \ 4.6 \times 10^{-3}-2.5 \times 10^{-3}\\\\ \text{(c)}\ \ \left(4.5 \times 10^{6}\right) \times\left(1.5 \times 10^{-2}\right)\\\\ \text{(d)}\ \ \displaystyle \frac{7.6 \times 10^{5}}{1.9 \times 10^{-2}} \end{array}$

    Show/Hide Solution

    $\begin{array}{l} \text{(a)}\;\;\ 2.3\times {{10}^{2}}+1.7\times {{10}^{2}}\\ \ \ \ =(2.3+1.7)\times {{10}^{2}}\\ \ \ \ =4.0\times {{10}^{2}}\\\\ \text{(b)}\;\;\ 4.6\times {{10}^{{-3}}}-2.5\times {{10}^{{-3}}}\\ \ \ \ =(4.6-2.5)\times {{10}^{{-3}}}\\ \ \ \ =2.1\times {{10}^{{-3}}}\\\\ \text{(c)}\;\;\;\left( {4.5\times {{{10}}^{6}}} \right)\times \left( {1.5\times {{{10}}^{{-2}}}} \right)\\ \ \ \ =(4.5\times 1.5)\times {{10}^{{6-2}}}\\ \ \ \ =6.8\times {{10}^{4}}\\\\ \text{(d)}\;\;\;\displaystyle \frac{{7.6\times {{{10}}^{5}}}}{{1.9\times {{{10}}^{{-2}}}}}\\ \ \ \ =4.0\times {{10}^{{5+2}}}\\ \ \ \ =4.0\times {{10}^{7}} \end{array}$

    5.           Compute using scientific notation.

                  $\begin{array}{l} \text{(a)}\ \ \displaystyle \frac{2.5 \times 10^{2}}{0.25 \times 0.002}\\\\ \text{(b)}\ \ \displaystyle \frac{33,000,000 \times 0.4}{1.1 \times 30}\\\\ \text{(c)}\ \ \displaystyle \frac{50 \times 0.014 \times 0.30}{10500}\\\\ \text{(d)}\ \ \displaystyle \frac{7000 \times 80 \times 300}{400} \end{array}$

    Show/Hide Solution

    $\begin{array}{l}\text{(a)}\;\;\ \displaystyle \frac{{2.5\times {{{10}}^{2}}}}{{0.25\times 0.002}}\\\ \ \ =\displaystyle \frac{{2.5\times {{{10}}^{2}}}}{{2.5\times {{{10}}^{{-1}}}\times 2.0\times {{{10}}^{{-3}}}}}\\\ \ \ =\displaystyle \frac{1}{2}\times {{10}^{{2+1+3}}}\\\ \ \ =0.5\times {{10}^{6}}\\\ \ \ =5.0\times {{10}^{5}}\\\\\text{(b)}\;\;\ \displaystyle \frac{{33,000,000\times 0.4}}{{1.1\times 30}}\\\ \ \ =\displaystyle \frac{{3.3\times {{{10}}^{7}}\times 4.0\times {{{10}}^{{-1}}}}}{{1.1\times 3.0\times {{{10}}^{1}}}}\\\ \ \ =\displaystyle \frac{{3.3\times 4.0}}{{3.3}}\times {{10}^{{7-1-1}}}\\\ \ \ =4.0\times {{10}^{5}}\\\\\text{(c)}\;\;\;\displaystyle \frac{{50\times 0.014\times 0.30}}{{10500}}\\\ \ \ =\displaystyle \frac{{5.0\times {{{10}}^{1}}\times 1.4\times {{{10}}^{{-2}}}\times 3.0\times {{{10}}^{{-1}}}}}{{1.05\times {{{10}}^{4}}}}\\\ \ \ =\displaystyle \frac{{5.0\times 1.4\times 3.0}}{{1.05}}\times {{10}^{{1-2-1-4}}}\\\ \ \ =\displaystyle \frac{{2.1\times 10}}{{1.05}}\times {{10}^{{-6}}}\\\ \ \ =2\times {{10}^{{-5}}}\\\\\text{(d)}\;\;\;\displaystyle \frac{{7000\times 80\times 300}}{{400}}\\\ \ \ =\displaystyle \frac{{7\times {{{10}}^{3}}\times 8\times {{{10}}^{1}}\times 3\times {{{10}}^{2}}}}{{4\times {{{10}}^{2}}}}\\\ \ \ =\displaystyle \frac{{7\times 8\times 3}}{4}\times {{10}^{{3+2+1-2}}}\\\ \ \ =42\times {{10}^{4}}\\\ \ \ =4.2\times {{10}^{5}}\end{array}$