9th Class Math

Solution:

Apply the Quotient Law: log_b m - log_b n = log_b (m / n).

Since log_b b = 1:

Solution:

Apply the Product Law: log_b m + log_b n = log_b (m × n).

Since 128 = 2⁷:

Apply the Power Law: log_b (m^k) = k log_b m:

Solution:

1. Apply the Power Law to the first term: k log_b m = log_b (m^k).

Since 8 = 2³, we get 8^1/3 = (2³)^1/3 = 2:

2. Now substitute back and apply the Quotient Law:

Since 19 = 13² = 3^-2:

Solution:

Note: When no base is written, common logarithm base 10 is assumed (log = log₁₀).

1. Apply the Power Law to the first term:

2. Apply the Product Law:

Since 100 = 10²:

Solution:

1. Simplify the first term: 64 = 4³.

2. Simplify the second term: 25 = 5².

3. Add the values together:

Solution:

Apply the Product Law:

Since 12 × 0.25 = 12 × 14 = 3:

Solution:

1. Apply the Power Law:

2. Apply the Product Law:

Solution:

Apply the Quotient Law:

Solution:

1. Apply the Power Law to both terms:

2. Apply the Change of Base Law (log_y x = log xlog y):

Cancel out log 5:

Convert back to logarithmic base:

Solution:

1. Apply the Power Law to the first term:

2. Apply the Product Law:

Solution:

1. Apply the Power Law to the first term:

2. Apply the Quotient Law to the subtraction term:

3. Apply the Product Law to the addition term:

Solution:

Note: ln represents the natural logarithm (logarithm to base e).

1. Apply the Power Law to all terms:

2. Combine the addition terms using the Product Law:

3. Combine the subtraction term using the Quotient Law:

Solution:

Apply the Quotient Law:

Solution:

1. Write the square root as fractional exponent 12:

2. Apply the Power Law:

3. Apply the Product Law to expand product factors:

4. Express 8 as 2³, and apply the Power Law to both terms:

5. Distribute 12 across terms:

Solution:

1. Apply the Quotient Law:

2. Apply the Product Law to expand the first product term:

3. Apply the Power Law:

Solution:

1. Apply the Power Law:

2. Apply the Quotient Law:

3. Apply the Product Law to split xy:

4. Distribute the coefficient:

Solution:

1. Convert the cube root into exponent 13:

2. Apply the Power Law:

3. Apply the Product Law:

4. Express 16 as 2⁴, and apply the Power Law to both terms:

5. Distribute 13:

Solution:

1. Apply the Power Law:

2. Apply the Quotient Law:

3. Distribute the coefficient:

Solution:

1. Combine into a single logarithm using the Product Law (base 10):

2. Convert to exponential form (recall that log y = z ⇒ 10^z = y):

Solution:

1. Combine into a single logarithm using the Product Law:

2. Convert to exponential form:

Solution:

1. Express both bases in powers of 3: 81 = 3⁴ and 243 = 3⁵.

2. Simplify exponents:

3. Equate exponents:

Solution:

1. Express both sides in terms of base 27:

2. Substitute back into the equation:

3. Compare exponents:

Solution:

1. Convert to exponential form (base 10):

2. Solve for x:

Solution:

1. Apply the Quotient Law to combine the logarithms:

2. Convert to exponential form:

3. Cross multiply and solve:

Solution:

1. Let x = 3.68 × 4.215.234. Take logarithm on both sides:

2. Apply logarithm laws to expand:

3. Look up values in log tables:

• log 3.68 = 0.5658
• log 4.21 = 0.6243
• log 5.234 = 0.7188

4. Add and subtract these decimal values:

5. Take antilogarithm on both sides: x = antilog(0.4713). Since characteristic is 0, lookup mantissa .47 under row 1, difference 3:

Solution:

1. Let x = 4.67 × 2.11 × 2.397. Take logarithm on both sides:

2. Look up log table values:

• log 4.67 = 0.6693
• log 2.11 = 0.3243
• log 2.397 = 0.3796

3. Add values:

4. Take antilog: x = antilog(1.3732). Characteristic = 1, Mantissa = .3732. Lookup row .37, column 3, difference 2:

Solution:

1. Let x = (20.46)² × 2.4122754.3. Round 2.4122 to 2.412. Take logarithm on both sides:

2. Expand using logarithm laws:

3. Look up table values:

• log 20.46 = 1.3109
• log 2.412 = 0.3824
• log 754.3 = 2.8776

4. Calculate:

5. Take antilog: x = antilog(0.1266). Characteristic = 0, Mantissa = .1266. Lookup row .12, column 6, difference 6:

Solution:

1. Let x = ³√{9.364} × 21.643.21. Take logarithm on both sides:

2. Expand using laws:

3. Look up log table values:

• log 9.364 = 0.9715
• log 21.64 = 1.3353
• log 3.21 = 0.5065

4. Substitute and compute:

5. Take antilog: x = antilog(1.1526). Characteristic = 1, Mantissa = .1526. Lookup row .15, column 2, difference 6:

Solution:

Given values: A = 10,000 and A₀ = 10.

Substitute these values into the formula:

Since 1000 = 10³:

Apply the Power Law:

Since log₁₀ 10 = 1:

Answer: The magnitude of the earthquake is 3 on the Richter scale.

Solution:

1. Set up the equation: The investment doubles when the value y = Rs. 200,000.

2. Divide both sides by 100,000:

3. Take logarithm on both sides:

4. Apply the Power Law:

5. Substitute log table values (log 2 = 0.3010, log 1.05 = 0.0212):

6. Rounding to the nearest whole year:

Answer: The investment will double after approximately 14 years.

Solution:

Given values: T₀ = 20, h = 500.

1. Substitute values into the formula:

2. Take logarithm on both sides:

3. Apply logarithm laws:

4. Find logarithms using tables. For log 0.97, characteristic is -1 (or 1) and mantissa is 0.9867:

5. Take antilogarithm: T = antilog(1.2345). Characteristic = 1, Mantissa = .2345. Lookup row .23, column 4, difference 5:

Answer: The temperature at an altitude of 500 metres will be approximately 17.16°C.