Logarithms??? Help!?

Question

1)

log(x+3)-log(2x-1)=2

2)

log4+log2x=log24

3)

logx-3 log5=2log10

Answer 1

1).

Method 1 :

The equation is log (x + 3) - log(2x - 1) = 2.

Apply quotient property logarithm : .

log [(x + 3)/(2x - 1)] = 2

log [(x + 3)/(2x - 1)] = 1 + 1

log [(x + 3)/(2x - 1)] = log(10) + log(10)

Apply product property of logarithm :log a + log b = log (ab ).

log [(x + 3)/(2x - 1)] = log(10 * 10)

Apply one - to - one property of logarithm : log a = log b ⇔ a = b.

(x + 3)/(2x - 1) = 100

(x + 3) = 100(2x - 1)

x + 3 = 200x - 100

199x = 103

⇒ x = 103/199 = 0.5175.

Solution of the equation is x = 0.5175.

Method 2 :

The equation is log (x + 3) - log(2x - 1) = 2.

Apply quotient property logarithm : .

log [(x + 3)/(2x - 1)] = 2

Apply exponential formula: log base e (x ) = a ⇔ x = e ^a.

[(x + 3)/(2x - 1)] = e^2

[(x + 3)/(2x - 1)] = 7.389

x + 3 = 7.389(2x - 1)

x + 3 = 14.778x - 7.389

14.778x - x = 3 + 7.389 = 10389

13.778x = 10.389

⇒ x = 10.389/13.778 = 0.75403.

Solution of the equation is x = 0.75403.

Answer 2

2).

The equation is log 4 + log(2x) = log 24.

log (2x) = log 24 - log 4

Apply quotient property logarithm : .

log (2x) = log (24/4) = log (6)

Apply one - to - one property of logarithm : log a = log b ⇔ a = b.

2x = 6

⇒ x = 6/2 = 3.

The solution of the equation is x = 3.

Answer 3

3).

The equation is log (x) - 3 log 5 = 2 log 10.

log (x) = 2 log 10 + 3 log 5.

Apply power property of logarithm : n log a = log a ^n .

log (x) = log (10)^2 + log (5)^3

Apply product property of logarithm :log a + log b = log (ab ).

log (x) = log (100 * 125) = log (12500)

Apply one - to - one property of logarithm : log a = log b ⇔ a = b.

x = 12500.

The solution of the equation is x = 12500.