What is x if lg(10x+4)=lg(4x-2) ?

First, we'll have to impose constraints of existance of
logarithms.

The first constraint:
10x+4>0

We'll divide by 2 both
sides:

5x+2>0

5x>-2

We'll
divide by 5 both
sides:

x>-2/5

The
second constraint: 4x-2>0

We'll divide by 2 boh
sides:

2x - 1>0

We'll
add 1 both
sides:

2x>1

We'll
divide by 2 both
sides:

x>1/2

The values
of x which satisfy both constraints belong to the interval (1/2 ,
+inf.)

Since the bases of logarithms are matching, we'll
solve the equation, using the one to one property of
logarithms:

10x + 4 = 4x -
2

We'll factorize by
2:

2(5x+2) = 2(2x-1)

We'll
divide by 2 both sides:

5x+2 =
2x-1

We'll subtract 2x both
sides:

5x - 2x + 2 = -1

3x + 2
= -1

We'll subtract 2 both
sides:

3x = -3

We'll divide by
3:

x = -1 <
1/2

The x value doesn't belong to the
interval of admissible values, so, the equation has no
solution.