Solve Log Equation: log7x + log(x+1) - log7 = log2x - logx

Question

$\log7x+\log(x+1)-\log7=\log2x-\log x$

$?=x$

00:00 Solve

00:07 Find the domain

00:22 This is the domain

00:40 We'll use the formula for logical addition, we'll get the log of their product

00:47 We'll use the formula for logical subtraction, we'll get the log of their quotient

01:06 We'll use these formulas in our exercise, and we'll get this log

01:36 Simplify what we can

02:01 Isolate X

02:16 We'll use the trinomial to find possible solutions

02:36 Check the domain

02:40 And this is the solution to the question

Defined domain

$x>0$

$x+1>0$

$x>-1$

$\log7x+\log\left(x+1\right)-\log7=\log2x-\log x$

$\log\frac{7x\cdot\left(x+1\right)}{7}=\log\frac{2x}{x}$

We reduce by: $7$ and by $X$

$x\left(x+1\right)=2$

$x^2+x-2=0$

$\left(x+2\right)\left(x-1\right)=0$

$x+2=0$

$x=-2$

Undefined domain $x>0$

$x-1=0$

$x=1$

Defined domain

$1$