Logarithm

Prepared for Gina. L

1)  A^B = C   ---->   B =log_A(C)

Proof

A^B = C

log A^B = log C  -- take log on both sides

B log A = log C 

B = log C / log A

B =log _A(C)

2)  log_b(mⁿ) = n · log_b(m)

example 1.1

5 log₅(27)

= 5 log₅(3³)

= (5 × 3) log₅(3)

= 15 log₅(3)

3)  log_b(mn) = log_b(m) + log_b(n)

example 2.1

log₂(20) 

= log₂(4 × 5) 

= log₂(4) + log₂(5) 

= log₂(2²) + log₂(5)

= 2 log₂(2) + log₂(5)

= 2 + log₂(5)

*Notice: log_A(A) = logA / logA = 1

4)  log_b(^m/_n) = log_b(m) – log_b(n)

example 3.1

log₂(15/4)

=  log₂(15) – log₂(4)

=  log₂(15) – log₂(2²)

=  log₂(15) – 2log₂(2)

=  log₂(15) – 2

5)  log_a(m) = 1 / log_m(a) 

example 4.1

log₂(9)

= 1 / log₉(2)

example 4.2

log₁₂(2) × log₂(144)

=  [1 / log₂(12)] × log₂(144)

= log₂(144) / log₂(12)

= log₁₂(144)

= log₁₂(12²)

= 2log₁₂(12)

= 2

6) For base = e, 2 and 10

log_e(m) = ln (m)

log₂(m) = lg (m)

log₁₀(m) = log (m)

note 5.1

Mathematical Constant e = 2.71828.... (irrational)

7) Differentiation

d/dx log_a(x) = (1/x) × (1/lna)

example 6.1

d/dx log₃(x)

= (1/x) × (1/ln3)

= 1/(x ln3)

d/dx log_a(g(x)) = (1/g(x)) × (1/lna) × g'(x)

example 6.2

d/dx log₅(g(x)),  given g(x) = 2x²

d/dx log₅(g(x))

= d/dx log₅(2x²)

= 1/(2x²) × 1/ln5 × d/dx (2x²)

= 1/(2x²) × 1/ln5 × 4x

⁼ (4x)/(2x² ln5)

d/dx ln(x) = 1/x

example 6.3

d/dx ln³(4x⁵)

= d/dx (ln(4x⁵))³  

= 3 (ln(4x⁵))^3-1 d/dx ln(4x⁵)

= 3 (ln(4x⁵))² (1/(4x⁵)) d/dx (4x⁵)

= 3 (ln(4x⁵))² (1/(4x⁵)) (20x⁴)

P.S.

Notice the following two equations are NOT the same.

ln³(m) = ln (m) × ln (m) × ln (m)

ln(m³) = ln (m × m × m) = 3 ln(m)

As well as these two equations.

Sin²(θ) = Sin (θ) × Sin (θ)

Sin(θ²) = Sin (θ × θ)

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written by : Kevin T (Mar 8, 2010)