logarthimic help

Question

4)

logx-2 log5x=2 

5)

5 logx +2 log3-3log2=4 

6)

log4-log(x+3)=log8 

 

Answer 1

4).

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

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

log x - log(5x)^2 = 2

Apply quotient property logarithm : .

log (x/(25x^2)) = 2

log (1/(25x)) = 2

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

1/(25x) = e^2

x = 1/(25e^2) = 1/(25 * 7.389) = 1/184.72

⇒ x = 0.00541.

The solution of the equation is x = 0.00541.

5).

The equation is 5 log (x) + 2 log 3 - 3 log 2 = 4.

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

log x^5 + log (3)^2 - log (2)^3 = 4

Apply quotient property logarithm : .

log x^5 + log(9/8) = 4

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

log [ x^5 * (9/8)] = 4

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

9x^5/8 = e^4

x^5 = 8e^4/9

x = [8e^4/9]^(1/5)

⇒ x = 2.1737.

The so;ution of the equation is x = 2.1737.

6).

The equation is log 4 - log (x + 3) = log 8.

log (x + 3) = log 4 - log 8

Apply quotient property logarithm : .

log (x + 3) = log (4/8) = log (1/2)

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

x + 3 = 1/2

2(x + 3) = 1

2x + 6 = 1

2x = 1 - 6 = - 5

⇒ x = - 5/2 = - 2.5.

The solution of the equation is x = - 2.5.