Solving a logarithmic equation where we have more than one log on the side. Whenever we have a log of equation, where we have more than one log, what we have to do is condense them down together to make a single logarithm. Using our properties of logs, put them back together to get a single log on whatever side has more than one.
For this one here our left side has two logarithms, what we need to do is condense them back down to a single one by subtraction which means that we are dividing when we put it back together, leaving us with x plus 5 over x minus 1 is equal to log base 7 of 3. Now we have the log base 7 of something equals the log base 7 of another thing. Those two things have to be the same, and just knowing that whatever we’re taking the log of has to be the same number so we just drop our logs out and we’re ending up with x plus 5 over x minus 1 is equal to 3.
From here, solving a simple equation, cross multiply, x plus 5 is equal to 3x minus 3. I skipped a step in here I just distributed that 3 in hopefully you can see it. Now we solve for x, let’s take all the x's to one side, everything else to the other side, add 3 over, subtract x and we end up with x is equal to 4.
Whenever we’re solving a logarithmic equation what we have to do is make sure that we can actually take this number and plug it into all of our original statements. So x is equal to 4, we can plug that in we get 9, that’s okay. Plug it in here, 4 minus 1 that’s 3, that’s okay and this log is already 3. So if we ever got a log of a negative number that would tell us that this number wouldn’t actually work, we’d have no solution. In this case we can take 4, we can plug it into all of our logs so we have a solution. Taking our log of equation, condensing everything down to a single log and then we would any other log.