Consider the equation that we have on the screen. 16 raised to the log base 4 of x minus 2 plus 10 log x plus 1 minus 4 log base 4 of 5 is equal to 0. What is the value of x in this equation? For those of you who want to try this problem, feel free to pause the video and work on it.
So what can we do to simplify this expression? Well, here's one of the properties of logs that you want to be familiar with. For example, let's say if you have 7 log base 7 of 9. It's important to understand that this is simply equal to 9. Here's why.
If you take the log of both sides, you'll have the log of 7, which is raised to log base 7 of 9. And on this side, you'll have log 9. Now, property of logarithms allows you to take the exponent and move it to the front. So you're going to have log base 7 of 9 times what we have here, log 7, and that's going to equal log 9. Next, you could use the change of base formula. Log base 7 of 9 is equal to log 9 over log 7. And when you multiply that by log 7, the log 7s will cancel. And so you can see that log 9 equals log 9. So this expression works.
So what that means is that we could simplify this expression here. 4 raised to the log base 4 of 5 is simply 5. Now what about this expression? What is the base if you don't see the base of a log?
If you don't see a base, always assume that it's base 10. So 10 log base 10 of x plus 1 is simply x plus 1. Now for the next one, we need to do some work. Because 16 and 4 don't match. However, we could change 16 into base 4. Because we know that 4 squared or 4 times 4 is 16. So let's replace 16 with 4 squared.
So we have 4 squared raised to the log base 4. of x minus 2. Now let's talk about that. x raised to the third power raised to the fourth power. What is that equal to? When you raise one exponent to another exponent, you need to multiply the two exponents. So this is equal to x to the 12th.
It's also equal to x to the fourth raised to the third, because 3 times 4 and 4 times 3 They're the same. But what I'm trying to help you see is that we could switch the 3 and 4. Because we could switch the 3 and 4, we could therefore switch the 2 and the log. So instead of writing 4 squared raised to log base 4 x minus 2, we can write 4 raised to the log base 4 x minus 2, and then all of that squared.
So all we did, we just swapped. those two exponents, which we can do because 3 times 4 is the same as 4 times 3. We don't need the parentheses around the x plus 1 at this point. Now, this we could simplify.
4 raised to the log base 4 of x minus 2. The 4s cancel, and we're just going to get x minus 2. So we have x minus 2 squared plus x plus 1 minus 5. 1 minus 5 is negative 4. So that's what we now have at this point. Now what we're going to do at this point is we're going to FOIL this expression. x minus 2 squared can be written as x minus 2 times x minus 2. So we have x times x, that's x squared.
x times negative 2. Negative 2x. Here we have another negative 2x. And then negative 2 times negative 2 is positive 4. So now we could cancel 4 and negative 4. Negative 2x minus 2x is negative 4x. And negative 4x plus x, that's negative 3x.
Now our next step is to factor out the GCF, which is x. And so we're going to have x times x minus 3. And then we can set each factor equal to 0. So adding 3 to both sides here will give us this solution. So we have two possible answers.
x can equal 0 or 3. But now, do both solutions work? Are both of them acceptable? Well, we can't plug in 0 into this expression because we'll have log negative 2, and you can't have a negative number inside of a log.
So, we can get rid of that solution. Now, just to be on the safe side, let's make sure that this solution works. So if we plug in 3, we're going to have 16 log base 4, 3 minus 2 plus 10 log base 3 plus 1, and then this is just 5. So 3 minus 2 is 1. 3 plus 1 is 4. Now, log of 1, regardless of what the base is, is 0. 4 to the 0 power is 1. So we're going to have 16 to the 0. This is going to be 4 minus 5 equals 0. Anything raised to the 0 power is 1. So 16 to the 0 power is 1. And 1 plus 4 is 5. 5 minus 5 is 0. And so the left side is equal to the right side, which means this is indeed the correct answer.