Solve the following problems involving logarithms: a) Find log(6.02214076 x 1023). b) If log x = -1.3649, find i) the characteristic and ii) the mantissa. c) Find the antilog of -1.3649.

Question

Asked on April 8, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-08T13:04:34.160Z

Here are the solutions to the problems: 11. Logarithms a) Use the calculator to find the logarithm of the constant 6.02214076 × 10^23. Let N = 6.02214076 × 10^23. We need to find _10 N. Using the logarithm property (AB) = A + B: _10 N = _10(6.02214076) + _10(10^23) _10 N = _10(6.02214076) + 23 Using a calculator: _10(6.02214076) ≈ 0.779750 So, _10 N ≈ 0.779750 + 23 _10 N ≈ 23.779750 The logarithm of the constant is 23.779750. b) Abdul works out the logarithm of a number as -1.3649. Determine the: i) Characteristics in the logarithm: To find the characteristic, we express the logarithm in the form characteristic + mantissa, where the mantissa is positive. -1.3649 = -1 - 0.3649 To make the mantissa positive, we add and subtract 1: -1 - 1 + 1 - 0.3649 = -2 + (1 - 0.3649) -2 + 0.6351 The characteristic is the integer part, which is -2. This is often written as 2. The characteristic is -2. ii) Mantissa in the logarithm: From the previous step, the mantissa is the positive decimal part. The mantissa is 0.6351. c) Hence use tables of antilogarithms to determine the value of 10^-1.3649. We need to find the antilogarithm of -1.3649. From part (b), we know that -1.3649 = 2.6351. To find the antilog of 2.6351: 1. Look up the mantissa 0.6351 in the antilogarithm tables. Antilog of 0.635 is 4.315. The mean difference for 1 (in the row for 63) is 1. So, the antilog of 0.6351 is 4.315 + 0.001 = 4.316. 2. The characteristic is 2, which means the decimal point should be placed such that there is one zero immediately after the decimal point before the first significant digit. (Alternatively, 10^2.6351 = 10^-2 × 10^0.6351 = 0.01 × 4.316). Therefore, the value is 0.04316. The value of 10^-1.3649 is 0.04316. 12. A grocery store has fruits in racks as shown in the table. Taking x to represent banana and y to represent orange, work out a simplified expression for the total number of fruits in the store. From the table: Number of bananas in Rack 1 = a Number of bananas in Rack 2 = a Total number of bananas = a + a = 2a. Number of oranges in Rack 1 = b Number of oranges in Rack 2 = b Total number of oranges = b + b = 2b. The total number of fruits in the store is the sum of total bananas and total oranges. Total fruits = 2a + 2b. This expression can be simplified by factoring out the common factor of 2. Total fruits = 2(a+b). The simplified expression for the total number of fruits is 2(a+b). 13. A carpenter at Gikomba furniture shop is making a rectangular table top. The length is (x+5) cm and the width is (x-2) cm. Form a simplified expression for the area of the table top. The area of a rectangle is given by the formula A = length × width. Given: Length L = (x+5) cm Width W = (x-2) cm Substitute these into the area formula: A = (x+5)(x-2) Expand the expression by multiplying the terms: A = x(x-2) + 5(x

Accepted Answer · 2026-04-08T13:04:34.160Z

Here are the solutions to the problems: 11. Logarithms a) Use the calculator to find the logarithm of the constant 6.02214076 × 10^23. Let N = 6.02214076 × 10^23. We need to find _10 N. Using the logarithm property (AB) = A + B: _10 N = _10(6.02214076) + _10(10^23) _10 N = _10(6.02214076) + 23 Using a calculator: _10(6.02214076) ≈ 0.779750 So, _10 N ≈ 0.779750 + 23 _10 N ≈ 23.779750 The logarithm of the constant is 23.779750. b) Abdul works out the logarithm of a number as -1.3649. Determine the: i) Characteristics in the logarithm: To find the characteristic, we express the logarithm in the form characteristic + mantissa, where the mantissa is positive. -1.3649 = -1 - 0.3649 To make the mantissa positive, we add and subtract 1: -1 - 1 + 1 - 0.3649 = -2 + (1 - 0.3649) -2 + 0.6351 The characteristic is the integer part, which is -2. This is often written as 2. The characteristic is -2. ii) Mantissa in the logarithm: From the previous step, the mantissa is the positive decimal part. The mantissa is 0.6351. c) Hence use tables of antilogarithms to determine the value of 10^-1.3649. We need to find the antilogarithm of -1.3649. From part (b), we know that -1.3649 = 2.6351. To find the antilog of 2.6351: 1. Look up the mantissa 0.6351 in the antilogarithm tables. Antilog of 0.635 is 4.315. The mean difference for 1 (in the row for 63) is 1. So, the antilog of 0.6351 is 4.315 + 0.001 = 4.316. 2. The characteristic is 2, which means the decimal point should be placed such that there is one zero immediately after the decimal point before the first significant digit. (Alternatively, 10^2.6351 = 10^-2 × 10^0.6351 = 0.01 × 4.316). Therefore, the value is 0.04316. The value of 10^-1.3649 is 0.04316. 12. A grocery store has fruits in racks as shown in the table. Taking x to represent banana and y to represent orange, work out a simplified expression for the total number of fruits in the store. From the table: Number of bananas in Rack 1 = a Number of bananas in Rack 2 = a Total number of bananas = a + a = 2a. Number of oranges in Rack 1 = b Number of oranges in Rack 2 = b Total number of oranges = b + b = 2b. The total number of fruits in the store is the sum of total bananas and total oranges. Total fruits = 2a + 2b. This expression can be simplified by factoring out the common factor of 2. Total fruits = 2(a+b). The simplified expression for the total number of fruits is 2(a+b). 13. A carpenter at Gikomba furniture shop is making a rectangular table top. The length is (x+5) cm and the width is (x-2) cm. Form a simplified expression for the area of the table top. The area of a rectangle is given by the formula A = length × width. Given: Length L = (x+5) cm Width W = (x-2) cm Substitute these into the area formula: A = (x+5)(x-2) Expand the expression by multiplying the terms: A = x(x-2) + 5(x

Solve the following problems involving logarithms: a) Find log(6.02214076 x 1023). b) If log x = -1.3649, find i) the characteristic and ii) the mantissa. c) Find the antilog of -1.3649.

Need help with your own homework?

More Mathematics Questions

Solve the following problems involving logarithms: a) Find log(6.02214076 x 1023). b) If log x = -1.3649, find i) the characteristic and ii) the mantissa. c) Find the antilog of -1.3649.

Need help with your own homework?

More Mathematics Questions