 So let's try and solve the equation log to base 4 x squared plus 3x minus log to base 4 x plus 5 equal to 1 So we need to express this as a single logarithmic expression So the first thing to notice here is that this is a difference of logs So we pull in our rules of logs and we see that if we have a difference of logs It's the same as the log of a quotient And so we can rewrite this as the log of x squared plus 3x over x plus 5 And we're still equal to 1 Remember definitions are the whole of mathematics. All else is commentary So every logarithmic equation corresponds to an exponential equation And so using the definition of logs gives us the equivalent equation 4 to power 1 Equals x squared plus 3x over x plus 5 And we can simplify this because we know that 4 to power 1 is equal to 4 And if it's not written down it didn't happen. So we'll record what we actually did over here on the left hand side And now we have a rational equation So we can start to simplify a rational equation by Multiplying by the denominator to eliminate the fraction. So our denominator is x plus 5 We'll multiply the left hand side by x plus 5 and because we have an equation We'll do the same thing to the right hand side And now we can simplify this Denominator factor of x plus 5 and this new factor of x plus 5 Well, they're common factors so we can remove them both leaving us with x squared plus 3x On the right hand side we can expand for times x plus 5 And now we have a quadratic equation So let's get all of our terms onto the one side or the other our x squared is already on the left So let's get rid of all the terms from the right hand side So we want to subtract 4x and subtract 20. We've got to do the same thing to both sides And we get our quadratic equation in a kind of gentle universe. We know that this was guaranteed to factor But one we don't live in that universe and two even if this was guaranteed to factor We usually have to go through a lot of trial and error to find the solutions So we're not going to waste our time factoring. We're going to instead solve using the quadratic formula Which gives us the solutions x equals 5 and x equals negative 4 Now if you don't want to use the quadratic formula because it's too easy to apply and always gives you an answer You could try to solve this by factoring And if it's a slow Friday night and you have nothing better to do and you've binge watched everything you can on Netflix Then solving by factoring is a great idea to occupy your time But solving using the quadratic formula always works and is always faster So there's no obvious reason why you do anything else Now it's important to remember that we should always check our solutions And this is true in general But it's especially true when we deal with things like equations that involve logs or square roots or exponents or Fractions or well really pretty much anything Here our concern is that since we're taking logs We can't take the log of a non-positive number. So we check our two solutions We check x equals minus 4 in the original equation and ask our self self Is it true that the log to base 4 x squared plus 3x minus the log to base 4 of x plus 5 is equal to 1 So we'll substitute in our values of x We'll do a little bit of arithmetic Simplification and we do need to know the log to base 4 of 4 and the log to base 4 of 1 Fortunately, we know the definitions because definitions are the whole of mathematics all else is commentary And so the log to base 4 of 4 is something So according to our definition 4 to power b should give us 4 And so we sit and stare at this equation and after a moment We realize that if b is equal to 1 this will be a true statement so b equals 1 and log to base 4 of 4 will be equal to 1 Similarly, we need to know the log to base 4 of 1 And so our definition says that whatever the log to base 4 of 1 is We know that 4 to that power gives us 1 And we stare at our equation and we remember that if I take a number and raise it to the 0 power I get 1 so this exponent b must be 0 Equals means replaceable so log to base 4 of 1 must be 0 And so we have to ask ourselves is it true that 1 minus 0 is equal to 1 And it is so x equals negative 4 is a solution We need to go through the same process with our other possible solution x equals 5 So we'll substitute x equals 5 into the original equation Do a little arithmetic Now at this point strictly speaking we should find the log to base 4 of 40 and the log to base 4 of 10 But because both 40 and 10 are Positive numbers we know they have a log so we can combine those logs because this is the Difference of two logs. We could rewrite it as the log of the quotient 40 over 10 And we can simplify this a little bit 40 divided by 10 is 4 And remember paper is cheap So of course you wrote down our earlier discovery that log to base 4 of 4 was equal to 1 And so this is also a true statement So x equals 5 is also a solution and so we verify that both are solutions