please check my answers.

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

ln sqrt x+6 = 9 i got {e^9-6}

log (x + 4) = log (5x - 5)
i got: 9/4

log (x + 30) - log 2 = log (3x + 2)
i got: {58}

Please limit yourself to one question per post, and show your work

log (x + 4) = log (5x - 5)

As long as the bases of both logs are the same,

x + 4 = 5x - 5
4x = 9
x = 9/4
=======================
log (x + 30) - log 2 = log (3x + 2)
(x+30)/2 = (3x+2)
x + 30 = 6x + 4
5x = 26
x = 5.2

true or false, all rational numbers are integers. Explain

To solve logarithmic equations, we need to isolate the logarithmic expression on one side of the equation. Let's take a look at each equation step by step:

1. ln sqrt(x+6) = 9
To start, eliminate the square root by raising both sides to the power of e. This will cancel out the natural logarithm.
e^(ln sqrt(x+6)) = e^9
sqrt(x+6) = e^9
Now, square both sides of the equation to eliminate the square root.
(x+6) = (e^9)^2
(x+6) = e^18
Finally, subtract 6 from both sides to isolate x.
x = e^18 - 6

2. log(x + 4) = log(5x - 5)
In this equation, both sides have the same base, which is 10. Therefore, we can simply equate the arguments of the logarithms.
x + 4 = 5x - 5
Rearrange the equation by moving all the x terms to one side and the constants to the other side.
4 + 5 = 5x - x
9 = 4x
Simplify by dividing both sides by 4.
x = 9/4

3. log(x + 30) - log 2 = log(3x + 2)
Using the logarithmic property that states log(a) - log(b) = log(a/b), we can rewrite the equation as:
log((x + 30)/2) = log(3x + 2)
Now equate the arguments of the logarithms.
(x + 30)/2 = 3x + 2
First, let's eliminate the fraction by multiplying both sides by 2.
x + 30 = 6x + 4
Next, rearrange the equation.
6x - x = 30 - 4
5x = 26
Divide both sides by 5.
x = 26/5

Therefore, the solutions to each equation are:
1. x = e^18 - 6
2. x = 9/4
3. x = 26/5