FINDING LOGARITHMS WITHOUT USING TABLES. Example 1: log₁₀2 = 0.3010 and log₁₀3 = 0.4771. Evaluate log₁₀4.5 Solution: log₁₀4.5 =  log₁₀45/10 =  log₁₀9/2 = log₁₀9 -  log₁₀2  = log₁₀3² -  log₁₀2 = 2 x 0.4771 -  0.3010  = 0.6532 Example 2: if log₁₀2 = 0.30103, find without using logarithms tables the number of digit in 2⁶⁴  Solutions: 2⁶⁴ = y log₂y = 64 logy/log2 = 64 logy/ 0.30103 =64 64 x 0.30103 = logy 10¹⁹·²⁶⁵⁹ = 20 Example 3: Given that log₁₀5 = 0.6990 and log₁₀7 = 0.8451. Without using tables, find the value of log₁₀140. Hence, solve x⁰·⁹⁶⁴¹ = 140 Solution: log₁₀140 = log₁₀700/5 = log₁₀700 - log₁₀5 =  log₁₀ 7 x 100 - log₁₀5= log₁₀7 + log₁₀10  - log₁₀5  substitute the logs into the equation  = 0.8451 + 2 - 0.6990 = 2.1461 Hence,     x⁰·⁹⁶⁴¹ = log₁₀140 ⇒ take log of both side log₁₀x⁰·...

Any mathematic questions with equal to sign (=) is called equation logarithm equation are algebraic equation involving logarithms.       WORKED EXAMPLES Example 1: Solve the equation log₃x² = -2 Solution: 3⁻² = x² ⇒ from definition Add square root to both side √(3⁻²) = √(x)² 3⁻² ˣ ½ = x² ˣ ½ 3⁻¹ = x 1/3 = x Example 2: Calculate the value of log₂ᵃ if logₐ4 =1/2 Solution a^½ = 4 a^½ˣ² = 4² a = 16 Hence, log₂a = log₂16 = log₂2⁴  = 4log₂2 = 4 x 1 = 4 Example 3: log₈32 = y + 2 8⁽ʸ ⁺²⁾ = 32 2³⁽ʸ ⁺² ⁾=2⁵ 3(y + 2) =5 3y + 6 =5 3y = 5 -6 y =-1/3 Example 4: Find x if log₃2 + log₃⁽²ˣ ⁺ ³⁾ - log₃⁽ˣ ⁻¹⁾ = 2  Solution: log₃ (2) x (2x + 3)÷( x - 1) = 32 log₃ (2) x ( 2x +3) ÷ ( x - 1) =log₃3 x 2 (2)(2x +3)/(x -1) = 3 x 2 (4x + 6)/ (x - 1) = 6 6( x - 1) = (4x + 6) 6x - 6 = 4x + 6 6x - 4x = 6 + 6 2x = 12 x = 12/2  = 6 Example 5: log₄( a² + a + 10) - log₄(a² + a - 5) = 1/2,...