Exercise 2.3 — Class 9 Mathematics Solutions

Solution

1. Express the number in scientific notation.

   $$5287=5.287\times {10}^3$$

2. Determine the characteristic.

   The characteristic of $\log 5287$ is the power of $10$:

   $$\begin{array}{cc}\text{Characteristic}& =3\end{array}$$

Solution

1. Express the number in scientific notation.

   $$59.28=5.928\times {10}^1$$

2. Determine the characteristic.

   $$\begin{array}{cc}\text{Characteristic}& =1\end{array}$$

Solution

1. Express the number in scientific notation.

   $$0.0567=5.67\times {10}^{-2}$$

2. Determine the characteristic.

   Since the exponent of $10$ is $-2$:

   $$\begin{array}{cc}\text{Characteristic}& =\bar{2}(\text{or }-2)\end{array}$$

Solution

1. Express the number in scientific notation.

   $$234.7=2.347\times {10}^2$$

2. Determine the characteristic.

   $$\begin{array}{cc}\text{Characteristic}& =2\end{array}$$

Solution

1. Express the number in scientific notation.

   $$0.000049=4.9\times {10}^{-5}$$

2. Determine the characteristic.

   Since the exponent of $10$ is $-5$:

   $$\begin{array}{cc}\text{Characteristic}& =\bar{5}(\text{or }-5)\end{array}$$

Solution

1. Express the number in scientific notation.

   $$145000=1.45\times {10}^5$$

2. Determine the characteristic.

   $$\begin{array}{cc}\text{Characteristic}& =5\end{array}$$

Solution

1. Find the characteristic.

   $$43=4.3\times {10}^1⟹\text{Characteristic}=1$$

2. Find the mantissa from log tables.

   Looking at row $43$, column $0$:

   $$\text{Mantissa}=0.6335$$

3. Combine characteristic and mantissa.

   $$\begin{array}{cc}\log 43& =\text{Characteristic}+\text{Mantissa}\\ & =1+0.6335\\ & =1.6335\end{array}$$

Solution

1. Find the characteristic.

   $$579=5.79\times {10}^2⟹\text{Characteristic}=2$$

2. Find the mantissa from log tables.

   Looking at row $57$, column $9$:

   $$\text{Mantissa}=0.7627$$

3. Calculate $\log 579$.

   $$\begin{array}{cc}\log 579& =2+0.7627\\ & =2.7627\end{array}$$

Solution

1. Find the characteristic.

   $$1.982=1.982\times {10}^0⟹\text{Characteristic}=0$$

2. Find the mantissa.

   • Value in row $19$, column $8$ $=0.2967$
   • Mean difference for $2$ $=4$

   $$\begin{array}{cc}\text{Mantissa}& =0.2967+0.0004\\ & =0.2971\end{array}$$

3. Combine.

   $$\begin{array}{cc}\log 1.982& =0+0.2971\\ & =0.2971\end{array}$$

Solution

1. Find the characteristic.

   $$0.0876=8.76\times {10}^{-2}⟹\text{Characteristic}=\bar{2}$$

2. Find the mantissa.

   Looking at row $87$, column $6$:

   $$\text{Mantissa}=0.9425$$

3. Write the logarithm in bar notation.

   $$\begin{array}{cc}\log 0.0876& =\bar{2}.9425\end{array}$$

Solution

1. Find the characteristic.

   $$0.047=4.7\times {10}^{-2}⟹\text{Characteristic}=\bar{2}(\text{or }-2)$$

2. Find the mantissa.

   Looking at row $47$, column $0$ in the logarithm table:

   $$\text{Mantissa}=0.6721$$

3. Write the logarithm.

   Combining characteristic and mantissa in standard textbook bar notation:

   $$\begin{array}{cc}\log 0.047& =\bar{2}.6721\end{array}$$

   (Note: In pure negative decimal form as computed by calculators: $-2+0.6721=-1.3279$).

Solution

1. Find the characteristic.

   $$0.000354=3.54\times {10}^{-4}⟹\text{Characteristic}=\bar{4}(\text{or }-4)$$

2. Find the mantissa.

   Looking at row $35$, column $4$ in the logarithm table:

   $$\text{Mantissa}=0.5490$$

3. Write the logarithm.

   Combining characteristic and mantissa in standard textbook bar notation:

   $$\begin{array}{cc}\log 0.000354& =\bar{4}.5490\end{array}$$

   (Note: In pure negative decimal form as computed by calculators: $-4+0.5490=-3.4510$).

Solution

1. Determine the characteristic of $3177$.

   $$3177=3.177\times {10}^3⟹\text{Characteristic}=3$$

2. Identify the mantissa.

   Since the significant digits $3177$ are identical to $3.177$:

   $$\text{Mantissa}=0.5019$$

3. Combine characteristic and mantissa.

   $$\begin{array}{cc}\log 3177& =\text{Characteristic}+\text{Mantissa}\\ & =3+0.5019\\ & =3.5019\end{array}$$

Solution

1. Determine the characteristic of $31.77$.

   $$31.77=3.177\times {10}^1⟹\text{Characteristic}=1$$

2. Identify the mantissa.

   $$\text{Mantissa}=0.5019$$

3. Combine.

   $$\begin{array}{cc}\log 31.77& =1+0.5019\\ & =1.5019\end{array}$$

Solution

1. Determine the characteristic of $0.03177$.

   $$0.03177=3.177\times {10}^{-2}⟹\text{Characteristic}=\bar{2}(\text{or }-2)$$

2. Identify the mantissa.

   Since the significant digits $3177$ are identical to $3.177$, the mantissa remains unchanged:

   $$\text{Mantissa}=0.5019$$

3. Write the logarithm.

   $$\begin{array}{cc}\log 0.03177& =\bar{2}.5019\end{array}$$

   (Note: In pure negative decimal form: $-2+0.5019=-1.4981$).

Solution

1. Take antilogarithm of both sides.

   $$x=\text{antilog}(0.0065)$$

2. Identify characteristic and mantissa.

   • $\text{Characteristic }(c)=0$
   • $\text{Mantissa }(m)=0.0065$

3. Look up the antilog table.

   • Row $.00$, column $6$ $=1014$
   • Mean difference for $5$ $=1$

   $$\begin{array}{cc}\text{Digits}& =1014+1=1015\end{array}$$

   Since $c=0$, place the decimal point after $0+1=1$ digit:

   $$\begin{array}{cc}x& =1.015\end{array}$$

Solution

1. Take antilogarithm of both sides.

   $$x=\text{antilog}(1.1920)$$

2. Identify characteristic and mantissa.

   • $\text{Characteristic }(c)=1$
   • $\text{Mantissa }(m)=0.1920$

3. Look up the antilog table.

   In row $.19$, column $2$, the value is $1556$. Since $c=1$, place the decimal point after $1+1=2$ digits:

   $$\begin{array}{cc}x& =15.56\end{array}$$

Solution

1. Convert negative decimal to standard bar notation with positive mantissa.

   $$\begin{array}{cc}\log x& =-3.4340\\ & =-4+(4-3.4340)\\ & =-4+0.5660\\ & =\bar{4}.5660\end{array}$$

2. Identify characteristic and mantissa.

   • $\text{Characteristic }(c)=\bar{4}$
   • $\text{Mantissa }(m)=0.5660$

3. Look up the antilog table.

   In row $.56$, column $6$, the value is $3681$. Since $c=\bar{4}$, place $3$ zeros after the decimal point before the first significant digit:

   $$\begin{array}{cc}x& =0.0003681\end{array}$$

Solution

1. Convert negative decimal to standard bar notation.

   $$\begin{array}{cc}\log x& =-1.5726\\ & =-2+(2-1.5726)\\ & =-2+0.4274\\ & =\bar{2}.4274\end{array}$$

2. Identify characteristic and mantissa.

   • $\text{Characteristic }(c)=\bar{2}$
   • $\text{Mantissa }(m)=0.4274$

3. Look up the antilog table.

   • Row $.42$, column $7$ $=2673$
   • Mean difference for $4$ $=2$

   $$\begin{array}{cc}\text{Digits}& =2673+2=2675\end{array}$$

   Since $c=\bar{2}$, place $1$ zero after the decimal point:

   $$\begin{array}{cc}x& =0.02675\end{array}$$

Solution

1. Take antilogarithm of both sides.

   $$x=\text{antilog}(4.3561)$$

2. Identify characteristic and mantissa.

   • $\text{Characteristic }(c)=4$
   • $\text{Mantissa }(m)=0.3561$

3. Look up the antilog table.

   • Row $.35$, column $6=2270$
   • Mean difference for $1=1$

   $$\begin{array}{cc}\text{Digits}& =2270+1\\ & =2271\end{array}$$

   Since $c=4$, place the decimal point after $4+1=5$ digits:

   $$\begin{array}{cc}x& =\mathrm{22,710}\end{array}$$

   (Note: Using standard 4-figure antilogarithm tables the result is $\mathrm{22,710}$; calculator computation gives ${10}^{4.3561}\approx \mathrm{22,703.88}$).

Solution

1. Convert negative decimal to standard bar notation.

   $$\begin{array}{cc}\log x& =-2.0184\\ & =-3+(3-2.0184)\\ & =-3+0.9816\\ & =\bar{3}.9816\end{array}$$

2. Identify characteristic and mantissa.

   • $\text{Characteristic }(c)=\bar{3}$
   • $\text{Mantissa }(m)=0.9816$

3. Look up the antilog table.

   • Row $.98$, column $1$ $=9572$
   • Mean difference for $6$ $=13$

   $$\begin{array}{cc}\text{Digits}& =9572+13=9585\end{array}$$

   Since $c=\bar{3}$, place $2$ zeros after the decimal point:

   $$\begin{array}{cc}x& =0.009585\end{array}$$