Sorry I am having log troubles

Here might be a slightly easier approach

log₇ (x) + log₇ (x + 6) = 1  

Note that we can write 1 as  log₇(7)

And using a log property, we can write the left side as log₇ [ x * (x + 6)]  ....so we have

log₇ (x * (x + 6) ] =   log₇(7)       and we can get rid of the logs and solve this equation :

x * (x + 6)  = 7

x^2 + 6x  = 7

x^2 +6x - 7  = 0       factor

(x + 7) (x - 1)  =  0      and setting both factors to 0, we have that x = 1, or x = -7

We must reject the second solution because it requires taking the logs of negative numbers in the original equation, and this is not defined........so, the aswer is  .....   x = 1

     Solve for X:
(log(X))/(log(7))+(log(X+6))/(log(7)) = 1

Rewrite the left hand side by combining fractions. (log(X))/(log(7))+(log(X+6))/(log(7))  =  (log(X)+log(X+6))/(log(7)):
(log(X)+log(X+6))/(log(7)) = 1

Multiply both sides by log(7):
log(X)+log(X+6) = log(7)

log(X)+log(X+6) = log(X (X+6)):
log(X (X+6)) = log(7)

Cancel logarithms by taking exp of both sides:
X (X+6) = 7

Expand out terms of the left hand side:
X^2+6 X = 7

Add 9 to both sides:
X^2+6 X+9 = 16

Write the left hand side as a square:
(X+3)^2 = 16

Take the square root of both sides:
X+3 = 4 or X+3 = -4

Subtract 3 from both sides:
X = 1 or X+3 = -4

Subtract 3 from both sides:
X = 1 or X = -7

(log(X))/(log(7))+(log(X+6))/(log(7)) => (log(-7))/(log(7))+(log(6-7))/(log(7))  =  1+((2 i) pi)/(log(7)) ~~ 1.+3.22892 i:
So this solution is incorrect

(log(X))/(log(7))+(log(X+6))/(log(7)) => (log(1))/(log(7))+(log(6+1))/(log(7))  =  1:
So this solution is correct

The solution is:
Answer: | 
| X = 1                                   

why do you add 9 to both sides?

Did you read and understand the operation AFTER that?

Our guest added 9 to both sides to make the LHS a perfect square.

he/she really added      (6/2)^2   which is 9 :))

CPhill's way is easier but it is good to understand lots of different ways if you can.

Thanks CPhill and thanks Guest for your great answers :))