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 $10$:

   $$\begin{aligned} \text{Characteristic} &= 3 \end{aligned}$$

Solution

1. Express the number in scientific notation.

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

2. Determine the characteristic.

   $$\begin{aligned} \text{Characteristic} &= 1 \end{aligned}$$

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 $-2$:

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

Solution

1. Express the number in scientific notation.

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

2. Determine the characteristic.

   $$\begin{aligned} \text{Characteristic} &= 2 \end{aligned}$$

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 $-5$:

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

Solution

1. Express the number in scientific notation.

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

2. Determine the characteristic.

   $$\begin{aligned} \text{Characteristic} &= 5 \end{aligned}$$

Solution

1. Find the characteristic.

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

2. Find the mantissa from log tables.

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

   $$\text{Mantissa} = 0.6335$$

3. Combine characteristic and mantissa.

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

Solution

1. Find the characteristic.

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

2. Find the mantissa from log tables.

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

   $$\text{Mantissa} = 0.7627$$

3. Calculate $\log 579$.

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

Solution

1. Find the characteristic.

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

2. Find the mantissa.

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

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

3. Combine.

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

Solution

1. Find the characteristic.

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

2. Find the mantissa.

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

   $$\text{Mantissa} = 0.9425$$

3. Write the logarithm in bar notation.

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

Solution

1. Find the characteristic.

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

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:

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

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

Solution

1. Find the characteristic.

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

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:

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

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

Solution

1. Determine the characteristic of $3177$.

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

2. Identify the mantissa.

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

   $$\text{Mantissa} = 0.5019$$

3. Combine characteristic and mantissa.

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

Solution

1. Determine the characteristic of $31.77$.

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

2. Identify the mantissa.

   $$\text{Mantissa} = 0.5019$$

3. Combine.

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

Solution

1. Determine the characteristic of $0.03177$.

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

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.

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

   (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 $5$ $= 1$

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

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

   $$\begin{aligned} x &= 1.015 \end{aligned}$$

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:

   $$\begin{aligned} x &= 15.56 \end{aligned}$$

Solution

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

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

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:

   $$\begin{aligned} x &= 0.0003681 \end{aligned}$$

Solution

1. Convert negative decimal to standard bar notation.

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

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 $4$ $= 2$

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

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

   $$\begin{aligned} x &= 0.02675 \end{aligned}$$

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 $1 = 1$

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

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

   $$\begin{aligned} x &= 22{,}710 \end{aligned}$$

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

Solution

1. Convert negative decimal to standard bar notation.

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

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 $6$ $= 13$

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

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

   $$\begin{aligned} x &= 0.009585 \end{aligned}$$