Japanese| A Math Olympiad algebra question

Welcome to my channel. Let's go ahead and find the value of M. Now, first let's divide through by what? 4 to the power of what? M. 4 to the power of M. If we do that, we'll have 1 plus 6 over 4 to the power of M, right? Since they have the same power, you can actually place the two inside parentheses and raise them to the power of M. It is equal to the same thing here, 9 over 4 raised to the power of M. Okay, so we have 1 plus, let's reduce this to its lowest terms. When you divide this by two, you'll have three. When you divide this, you'll have two. So the same thing as 3 over 2 to the power of M is equal to the same thing here. This is the same thing as 3 squared over 2 squared, raised to the power of M. Alright, so this is 1 plus 3 over 2 to the power of M equals 3 over 2 squared raised to the power of M. Okay, so recall that 8 raised to the power of M, raised to the power of N, is the same as 8 raised to the power of N, raised to the power of M. Okay, so we can actually swap their positions. Okay, we can interchange their positions. And ifModification, we have 1 plus 3 over 2 to the power of M equals 3 over 2 to the power of M squared. Okay, now we have 3 over 2 to the power of M, 3 over 2 to the power of M. So let's give it an alphabet. So let's set 3 over 2 equal to Y. Alright, 3 over 2 to the power of M equals Y. So if we say that, that means this equation becomes 1 plus Y equals Y squared, right? Yeah, now let's rewrite this because we've just formed a quadratic equation. So if we rewrite this, we have Y squared; this will cross over to become minus Y, and this will cross over to become minus 1, which is equal to 0. Alright, now let's use the quadratic formula, which states that Y equals negative B plus or minus the square root of B squared minus 4AC, all divided by 2A. Okay, in this case, A equals the coefficient of Y squared, which is 1, B equals the coefficient of Y, which is minus 1, and C is the constant, which is minus 1. Okay, so if we substitute this into the equation, we get Y equals minus B squared, which is minus minus 1, resulting in plus 1. plus or minus the square root of B squared, which is minus 1 squared, giving us 1, minus 4 times, our A is 1 and our C is minus 1, divided by 2A, that is 2 times 1. Alright, so we have Y equal to 1 plus or minus the square root of 1, which gives us minus 4 and minus times minus becomes plus. So we have plus 4 divided by 2. This implies that Y is equal to 1 plus or minus the square root of 5, divided by 2. Okay, so we have two values of Y here. The first one is that your Y is equal to 1 plus the square root of 5 divided by 2, and the second one is that Y is equal to 1 minus the square root of 5 divided by 2. So recall that our Y is 3 over 2 to the power of M. So we have 3 over 2 to the power of M equals 1 plus the square root of 5 divided by 2. In this case, we have 3 over 2 to the power of M equals 1 minus the square root of 5 divided by 2. But on the left-hand side, we are going to obtain a negative solution here. And once this side is negative and we are half positive here, it's not possible. There is no solution here, so this place has no solution. Okay, that means you're going to focus on this side of the equation. So what do we do from here? We have 3 over 2 to the power of M is equal to 1 plus the square root of 5 divided by 2. Now let's introduce log, because that is the only option I'm seeing here. So if we say log (3/2 M) equals log of (1 plus the square root of 5) divided by 2. Okay, now let's continue from here. Let's continue. Okay, so this is what we have. We have log base 3 over 2 of M equals log of 1 plus the square root of 5 divided by 2. Okay, so we can bring this down, right? Because the logarithm of A to the power of B is the same as B times the logarithm of A. Okay, so we can actually reduce our M. So if we do that, we have M times log 3 over 2 equals what? Log of 1 plus the square root of 5, divided by 2. Alright, so let's divide both sides of this equation by log 3 over 2 divided by log 3 over 2. So if we do that, we cancel this, and we are left with M being equal Let me write it here. So what are we left with? M is equal to log of (1 plus square root of 5) divided by 2, all divided by log of 3 divided by 2. Now, recall the logarithmic rule that states log A divided by B is the same as log A minus log B. A minus log B. Okay, alright, so that means we can write this as M equals log what? 1 plus the square root of 5 minus log 2, that is the numerator, divided by log 3 minus log 2. Okay, so M equals log of 1 plus root 5 will give us what? Alright, the log of this using my calculator gave me 0.5100 minus log 2. Log 2 is approximately 0.3010 to four decimal places. Divided by log 3. Log 3 is 0.4771, and log 2, we have obtained that previously. So if we solve this, we will have 0.5100 minus 0.3010. That will give us 0.209 divided by 0.4771 minus 0.3010 divided by 0.1761. Okay, now this is equal to, well, M is equal to, we have approximately 1.1868. Approximately, this is what you will get if you press this on your calculator. Thank you so much for watching. Have a lovely day. Bye.