Japanese| Math Olympiad algebraic simplification.

Hi, let's quickly solve for the value of x. This is JJ Online Math Class, and you are highly welcome. Okay, the value of x is actually very easy to solve since we can convert everything to base 10. okay so we have 10 to the power of x plus 10 squared to the power of x is equal to 10 to the power of 3 to the power of x, okay, now I want us to recall the law of exponents or indices that states that a raised to the power of m raised to the power of n is equal to a to the power of m times n m in that is m times n, so that simply means that we can write it as m raised to the power of n. power of m and we are still going to get the same thing, okay, so that means that the position of the powers can be interchanged, so let's go ahead and do that. So if we do that, here we have that 10 to the power of x plus 10 x squared, excuse me, is equal to 10 Pornhub to the power of 3. Now let's turn to the power of x equals y, all right? If we make that statement, that means we have that y plus y squared equals y to the power of 3; now notice that we have just formed a cubic equation. So let's move everything to this side of the equation to arrange it all. if we do that we have y to the power of three plus two y we plus y squared we cross to become minus y squared plus y, we move to the other side to become minus y equals zero, so let's factorize y we have y squared minus y minus one equals zero; now, if we divide through by y we are going to have that y squared minus y minus one equals zero, okay? Now so we have a quadratic equation here, so let's solve this using the quadratic equation formula and the quadratic The equation formula states that this y is equal to minus b plus or minus the square root of b squared minus four a c all over two a. Now, in this case, in this case, I want you to note that a is the coefficient of y squared, which is one; b is the coefficient of y, which is minus one. and since c is a constant which is also minus one, so if we should substitute this into this equation y is equal to. Recall that we are not looking for the value of y, rather we are looking for the value of x, okay, so our y is equal to minus b, that is minus minus one, which is one, plus or minus the square root of b squared, which is minus one squared, which is one, minus four a c, which is four times one times minus one, which is minus four, okay, divided by two a, that is two times one, which is So we have that y is equal to one plus or minus the square root of five, all divided by two. All right. That means we now have two values for our y. The first value is y₁ equals one. plus root five over two, okay, this is five, and y two is equal to one minus root five over two now For this particular side, we are going to get a negative number here, right? So this place is not. going to give us a new root; it's going to give us a complex root; therefore, we are not going to Use this; we are going to ignore this now. We continue with this: that's y is equal to one plus root five over two. Now recall that our y is equal to 10 to the power of x, right? So we write that 10. to the power of x is equal to one plus root five over two. Now let's introduce logarithmic reading to both sides of this equation, if we do that, we have log 10 to the power of x is equal to log, that is log to base 10 for each of them, one plus root five over two. Now, using the power rule, the power rule that if you have your power, you can bring down the power to multiply your log, okay, that means we You can write this as x multiplied by log 10 equals log of one plus root five over two. log base 10 of 10 is one, so that simply means that x is log base 10 of what one plus root five over two, thank you so much for watching, and have a lovely day