Solve for x log(base6)(x+3)−log(base6)(x−5)=2.

I tried to factor out the log(base6) and got (x+3)^x/(x-5)=2, but I don't know how to solve that...

sorry it's log(base6)(x+3)−log(base6)(x−5)=2

the x in front of log isn't part of it

Oh no, 6 is not a number, it is an operator.

You can't factor out 6
(that would be doing something like
√8 - √3
= (√)(8-3) )

You have to use your rules of logs.

6(x+3) - 6(x-5) = 2
6 ( (x+3)/(x-5) ) = 2

which by definition is
(x+3)/(x-5) = 6^2 = 36
36x - 180 = x+3
35x = 183
x = 183/35

Oh no, log6 is not a number, it is an operator.

You can't factor out log6
(that would be doing something like
√8 - √3
= (√)(8-3) )

You have to use your rules of logs.

log6(x+3) - log6(x-5) = 2
log6 ( (x+3)/(x-5) ) = 2

which by definition is
(x+3)/(x-5) = 6^2 = 36
36x - 180 = x+3
35x = 183
x = 183/35

thank you for your explanation :)

To solve the equation log(base6)(x+3) - log(base6)(x-5) = 2, you can use the properties of logarithms.

Step 1: Combine the two logarithms using the quotient rule of logarithms. The quotient rule states that log(basea)(x) - log(basea)(y) = log(basea)(x/y).

So, applying the quotient rule, we have:
log(base6)((x+3)/(x-5)) = 2.

Step 2: Convert the equation into exponential form. In exponential form, log(basea)(x) = y can be rewritten as a^y = x.

In this case, we have:
6^2 = (x+3)/(x-5).

Step 3: Simplify and solve for x. Distribute the exponent to both terms in the numerator:
36 = (x+3)/(x-5).

Step 4: Multiply both sides of the equation by (x-5) to eliminate the fraction:
36(x-5) = x+3.

Step 5: Expand and solve for x. Distribute 36 to the terms in parentheses:
36x - 180 = x + 3.

Step 6: Collect like terms and isolate the variable:
36x - x = 3 + 180,
35x = 183.

Step 7: Divide both sides by 35 to solve for x:
x = 183/35.

So, the solution to the equation log(base6)(x+3) - log(base6)(x-5) = 2 is x = 183/35.