Solving Simple Logarithmic Equations - Problem 3
Solving a simple logarithmic equation; whenever we're solving a simple logarithmic equation, it's equation in log form what we need to do is put into exponential form. So for this one 1/4stays where it is. The base x comes up and around and we're left with x to the -2.
We haven't really seen much with our variable in the base 7 exponential and I just want to talk about this for a second. So if we have 5² is equal to 5 to the x, our bases are the same, our exponents have to be the same, we then are left with 2 is equal to x.
Similarly if we have x to the fourth is equal to 5 to the fourth, now our exponents are the same, so our bases also have to be the same, so this has to leave us with x is equal to 5. So in this one over here, what we have to do is a similar thing, we can't get our bases the same because we have a variable and a number, but what we can do is get our exponents the same.
So looking at that, here is a -2. Remember when we're dealing with negative we could always make it positive by flipping the fraction, flipping the inside or we can somehow make this as something to the -2. I know that 4 is 2², so this is the same thing as 1/2 squared is equal to x to -2. Now all we have to do is somehow incorporate this difference of negative, flip one of our term bases, we could flip the 1/2 or we can flip the x, it doesn't matter just one of them is going to have to some to either the top or the bottom, the opposite of where it is.
I'm going to flip the x just to make both of our exponents positive, so what we're left with is 1/2 squared is equal to 1 over x². So now we have something where our exponents are the same, it tells us our bases have to be the same as well, so we can drop our exponents and basically we're left with 1/2 equals 1 over x.
By similar logic as we did over here, if we have a fraction where our numerators are the same, the denominators have to be the same as well. So we have 1 over 1, those are equal so therefore 2 has to equal x, leaving us with x equals 2.
This one is a little bit more straight forward, you can just look at this and say what to the -2 is equal to 1/4, 2, that would work, but for a little bit more complicated numbers, you may not always be able to see it straight up, so this sort of process of what you're going to have to go through.