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.