A Harvard University interview math test| Exponential math question| solve for x

Hi everyone, let's solve for x. First, let's factor out the common term, which is 4 raised to the power of x. So we have 4 raised to the power of x. When this is divided by this, we get 1. plus when this is divided by this, we have another 1, plus when this is divided by this, we have another 1, is equal to 40. So we have 4 raised to the power of x multiplied by 3 equals 40. Now, let us divide both sides of this equation by 3, okay? So we have 4 raised to the power of x multiplied by 3 divided by 3 equals 40 divided by 3. Now 3 will cancel itself or 3 will divide itself to give us 1. 1 times 4 to the power of x is equal to 4 to the power of x, which is equal to 40 divided by 3. Now, since what we are looking for is the power, we are going to use long reading. That is the logarithm or natural logarithm used to bring it down and find the value of x. So we are going to take the logarithm of both sides of this equation. So we have log 4 raised to the power of x equal to log 40 divided by 3, okay? All right, so let me shift this in. Okay. Now, when you have log a raised to the power of b, it is equal to b multiplied by log a. That means we can write this as x multiplied by log 4. So we have x times log 4 equals log 40 divided by 3. Now we have x log 4 is equal to remember that log a divided by b is equal to log a divided by log a minus log b. Okay, so that means we can write this as log 40. minus log 3. All right. Now remember we are looking for the value of x. So let us divide both sides of this equation by log 4. Note that the main reason for solving this is actually to obtain the value of x. I know so many of us are capable of doing that. The interesting part is that I proved the answer we obtained is correct. Yeah, whatever we got for x, we had to substitute it back into this to prove that it's correct. And I'm telling you that is the most interesting part of this video. Don't miss it. Okay, so we divide both sides by log 4. Okay, we divide both sides by log 4. Now log 4 cancels itself, or log 4 we divide itself to give us 1, and 1 times x is equal to so we have log 40 divided by log 4 - log 3 / log 4 So x is equal to, we can express 40 as 8 times 5. So we'll write this as log 8 multiplied by 5 divided by log 4 minus log 3 divided by log 4 So x Oh, this is the wrong one, girl. Okay. So our x is equal to When you have log a multiplied by b is equal to log a. plus log b. Okay. Yeah, so we can write this as log 8 plus log 5. So we have log 8 plus log 5 divided by log 4 minus log 3 divided by log 4 Alright, if this is the first time you've seen us, click the subscription button and turn on the notification bell. so that you'll be notified anytime we upload a new video. We upload videos every day. So tune in, confident that you'll find our latest uploads worth watching. Yeah, thank you very much. We upload two to three times daily. Now give us a thumbs up if you're actually enjoying this video. So let's continue now x is equal to log 8 divided by log 4 plus log 5 divided by log 4 minus log 3 divided by log 4. So we have x equal to x, which can be expressed as 2 raised to the power of 3. Okay divided by log 4 can be expressed as 2 squared now plus log 5 divided by (log 2) squared minus log 3 divided by log 2 squared. Now remember that's where you have something like this. You can bring this down to multiply. log 2, right? So write this to be x equals 3 log 2 divided by this, we come down to log 2 minus log 5 divided by 2 log 2 minus log 3 divided by 2 log 2 All right Let's continue from here Okay So we now have that x is equal to This will divide itself to give us one. So we are left with 3 over 2. Minus what we have here is half. Multiplied by log 5 divided by log 2 minus what we have here, which is also half multiplied by log 3 divided by log 2 Now at this point, I want us to remember a law in log reading, and this law states that log a over log b Is equal to log base b of a. Okay Yeah, so that means that We can write x. Equal to 3 over 2 minus one half log₂ 5 minus one half Log base 2 of 3. I see something here. Notice that we have 2, 2, 2. That means they have the same denominator. Now, let's bring everything onto that Let's put that, then let's also share that as a common denominator. So we have x will be 3 minus log₂ 5 Minus is minus minus? Oh, I said that, so this is plus, sorry. So this is plus this is plus yeah plus plus plus. All right Now we have 3 plus log base 2 of 5 minus log 11 in base 2 Divided by 2, so I think this is as far as we can go. Now that we are trying to find the value of x without a calculator, let us prove that the answer we obtained is correct. So the question states that 4 to the power of x plus 4 to the power of x plus 4 to the power of x. Is equal to 40 and we simplified it further as 4 to the power of Sorry, as 3 multiplied by 4 to the power of x Okay is equal to 40. All right. Now, let's substitute the value of x here. So we have 3 into 4 to the power of now we have 3 plus log₂ 5 minus log base 2 of 3 Divided by 2, okay, is equal to 40. Now we can express 4. We can express 4 As 2 to the power of 2, you can express 4 as 2 to the power of 2. So we have this to be equal to 3 times 2 to the power of 2, closed brackets, 3 plus log₂ 5 - log₂(3) Divided by 2 equals 40. Alright Now remember that when you have a to the power of m raised to the power of n Is equal to a to the power of m times n. So that means that we can actually Multiply this by this, and when we do that, this will divide itself to give us one. And that means that what we have now, let me continue from here So what it means is that what we have now Is 3 2 to the power of 3 multiplied by log base 2 of 5 Minus log base 2 of 3, and is this equal to 40? Now remember that when you have a to the power of m plus n Is equal to a to the power of m multiplied by a to the power of n Also, when you have a to the power of m minus n is equal to a to the power of m. Divided by a to the power of n. All right so That means that when we see addition, we multiply, and when we see subtraction, we divide. So write this as 3 multiplied by 2 to the power of 3 because of multiplication, we add the exponents of 2 to the power of Log base 2 of 5, because of subtraction, we divide 2 to the power of log base 2 of 3. Okay, is this equal to 14? Let's see So what we have is 3 into now 2 to the power of 3 is 8 multiplied by there is one important A law of log reading that we need to discuss. Yes, this law states that a to the power of Log base a of b is equal to b A to the power of log base a of b, once this base and this base are equal, is equal to b. Alright, so let's see if we need to apply it here. Notice that this and this are the same. That means the entire thing here is 5, and this is the same, so the whole of this is just 3. Okay is equal to 14. Now. This is what we have here. Let me write it here This is what we have 3 brackets 8 multiplied by 5 divided by 3. Okay is equal to 40 now 3 divided itself to give us 1 times 8 times 5 is still 8 times 5 Is equal to 40 now? 8 times 5 is 40, right? So you have 40 is equal to 40, and that is an absolute truth. Thank you so much for watching. See you in my next video. Bye