Answer to Question #174551 in Algebra for Richard

Question

1.log₁₁(x²+3x) =log₁₁(3x+16)

2.In(4-3x) -In(7x) =In(11)

3.log(x) +log(x+12)=log(x-10)

4.In(x) =In(15-x) -In(x+1)

5.log₈(4x+1) =-1

6.log₆(3x) -log₆(x+5) =1

Answer 1

1.

$x^2+3x=3x+16, 3x+16>0$

$x^2=16, x>-\dfrac{16}{3}$

$x_1=-4, x_2=4$

2.

$\ln(4-3x)=\ln(77x), 0<x<\dfrac{4}{3}$

$4-3x=77x$

$x=0.05$

3.

$\log(x(x+12))=\log(x-10), x>10$

$x^2+12x=x-10$

$x^2+11x+10=0$

$(x+1)(x+10)=0$

Since $x<0,$ then there are no solutions.

There are no solutions.

4.

$\ln(x)+\ln(x+1)=\ln(15-x), 0<x<15$

$\ln(x(x+1))=\ln(15-x)$

$x^2+x=15-x$

$x^2+2x-15=0$

$(x+5)(x-3)=0$

Since $0<x<15,$ we take $x=3.$

Solution is $x=3.$

5.

$4x+1=8^{-1}$

$4x=-\dfrac{7}{8}$

$x=-\dfrac{7}{32}$

6.

$\log_6(3x)=\log_6(6(x+5)), x>0$

$3x=6(x+5)$

$x=2x+10$

Since $x=-10<0,$ then there are no solutions.

There are no solutions.