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secondary 3 | A Maths

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I am unsure of how I'm supposed to do logarithms

(a)(i)

log₂ √x

= log₂ (x^½)

= ½ log₂ (x)

since log₂ x = p,

½ log₂ (x)

= ½ p

(ii)

log₂ xy²

= log₂ x + log₂ y²

= log₂ x + 2 log₂ y

since log₂ x = p, log₂ y = q,

log₂ x + 2 log₂ y

= p + 2q

(iii)

log₄ (4x/y)

= log₂ (4x/y) / log₂ 4

= [ log₂ 4x - log₂ y ] / log₂ 2²

= [ log₂ 4 + log₂ x - log₂ y ] / 2 log₂ 2

since log₂ x = p, log₂ y = q, & log₂ 2 = 1,

[ log₂ 4 + log₂ x - log₂ y ] / 2 log₂ 2

= [ log₂ 2² + p - q ] / 2

= [2 log₂ 2 + p - q ] / 2

since log₂ 2 = 1,

= [2 + p - q ] / 2

= 1 + ½(p - q)

(b)

lg x + lg[5(x+1) - 2 = 0

since lg 100 = 2,

lg x + lg[5(x+1) - lg 100 = 0

lg [ x (5x + 5)] - lg 100 = 0

lg (5x²+ 5x) - lg 100 = 0

lg [(5x²+ 5x)/100] = 0

lg [(x²+ x)/20] = 0

since lg 1 = 0,

lg [(x²+ x)/20] = lg 1

(x²+ x)/20 = 1

x²+ x = 20

x² + x - 20 = 0

(x + 5)(x - 4) = 0

x = -5 or x = 4

...

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