Question

How do you solve `log(5x+2)=log(2x-5)`?

Answer 1

`x= -7/3`

Explanation:

Given `log(5x+2) = log(2x-5)` common log- base 10

Step 1: Raised it to exponent using the base 10

`10^(log5x+2) = 10^(log2x-5)`

Step 2: Simplify, since `10^logA= A`
`5x+2= 2x-5`

Step 3: Subtract `color(red) 2` and `color(blue)(2x)` to both side of the equation to get
`5x+2color(red)(-2)color(blue)(-2x)= 2x color(blue)(-2x)-5color(red)(-2)`
`3x= -7`

Step 4: Dive both side by 3
`(3x)/3 = -7/3 hArr x= -7/3`

Step 5: Check the solution

`log[(5*-7/3)+2] = log[(2*-7/3) -5]`
`log(-35/3 + 6/3) = log(-14/3 -15/3)`
`log(-29/3) = log(-29/3)`

Both side are equal, despite we can't take a log of a negative number due to domain restriction `log_b x = y, , x >0 , b>0`
`x= -7/3` , assuming a complex valued logarithm