Question

a) log 4x = 3log 2 + 4log 3
b) log (2x-4)= 2
c) 4log (3 - 2x) = -1
d) log (x + 1) + log x = log (X + 9)
e) log (x + 3) = log 2 - log (x + 2)​

Answer 1

a)

$log(4x) = 3 log(2) + 4 log(3) \\ log(4x) = log( {2}^{3} ) + log( {3}^{4} ) \\ log(4x) = log(8 \times 81) \\ 4x = 648 \\ x = 162$

b)

$log(2x - 4) = 2 \\ 2x - 4 = 10 ^{2} \\ 2x = 104 \\ x = 52$

c)

$4 log(3 - 2x) = - 1 \\ (3 - 2x) {}^{4} = 10 {}^{ - 1} \\ 3 - 2x = 10 {}^{ - \frac{1}{4} } \\ - 2x = \frac{1}{ \sqrt[4]{10} } - 3 \\ - 2x = \frac{ \sqrt[4]{1000} }{10} - 3 \\ x = \frac{30 - \sqrt[4]{1000} }{20}$

d)

$log(x + 1) + log(x) = log(x + 9) \\ log((x + 1)(x)) = log(x + 9) \\ x {}^{2} + x = x + 9 \\ x {}^{2} = 9 \\ x = 3$

Nota: No se cuenta el " -3 " porque un logaritmo negativo no esta definido.

e)

$log(x + 3) = log(2) - log(x + 2) \\ log(x + 3) = log( \frac{2}{x + 2} ) \\ x + 3 = \frac{2}{x + 2} \\ x {}^{2} + 5x + 6 = 2 \\ x {}^{2} + 5x + 4 = 0 \\ (x + 1)(x + 4) = 0 \\ x + 1 = 0 \: \: \:o \: \: \: x + 4 = 0 \\ x = - 1 \: \: \: o \: \: \: x = - 4 \\ x = - 1$

Aqui ocurre los mismo que en la d) ya que si X toma el valor de " -4 " la expresion del logaritmo seria negativa.