A math Olympiad exponential math question| Many failed this| Can you solve for x? Exponents.

Hi, let's solve for the value of x. First, we are going to remember or recall the law of exponents that state that 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, so we are going to apply this law here, okay, so when we do that We have 4 to the power of x plus 25 to the power of x is equal to 10 to the power of x. multiplied by 10 to the power of 1, which is still 10, so we are going to divide through by 10 to the power of x divided by 10 to the power of x divided by 10 to the power of x divided by 10 to the power of x, so this will divide this; and another law of exponents we is the one that states that a to the power of m divided by b to the Okay, so we are going to apply this law here and here, and when we do that, we have 4 divided by 10 raised to the power of x plus 25 divided by 10 raised to the power of x is equal to 10, okay. Now, here's what they have in common: 2. So let's reduce this to its lowest term; so when 2 divides 4 we have 2, and when 2 divides 10 we have 5, so this is the same thing as 2 over 5 to the power of x, all right, then plus here they have 5 in common As the common factor, sorry, so 25 divided by 5 gives us 5, and 10 divided by 5 gives us 2. raised to the power of x is equal to 10. Now I want us to recall that 1 over x is equal to x. to the power of minus 1, okay, so that simply means that if we take the inverse of this, we are going So that's why we'll write 2 divided by 5 raised to the power of x, all right, plus 2. divided by 5, I've taken the inverse of 5 over 2, so the power becomes minus x is equal to 10 Alright, now also note that x to the power of minus 1 is equal to 1 over x, just as I said. before, so that means to remove this negative sign, I need to introduce 1 over to this place Right, so that is exactly what I will do now. So I'll write 2 divided by 5 raised to the power of x. plus 1 divided by 2 over 5 raised to the power of x all right is equal to 10 now we are going to say Let (2/5) raised to the power of x be equal to y. All right, so if we do that, we have that's y. plus 1 over y equals 10, okay? All right, if this is the first time you're coming across Please subscribe to the channel and click the notification bell so I'll be notified. Whenever I upload a new video, make sure you don't miss any of our videos, okay? And please give us a thumbs up. If you're actually enjoying this video, thank you. So the first thing we need to do now is to Multiply through by y because I want to obtain a linear equation, or a quadratic equation. So in order to do that, I need to multiply this through by y, so I need a y multiplied by y. y squared right plus 1 over y multiplied by y will give me 1 is equal to 10 And I want you to watch to the end because this is not as easy as you think, okay? Watch to the end. And see how we obtained the value of x from this y. All right, so we have y squared plus 1 equals 10. All right then, that means that y squared, this 10y, will cross over and become minus 10y. plus 1 is equal to 0, all right. Now we are going to find the value of y using the quadratic equation formula, right? So the quadratic equation formula states that y is equal to negative b plus or minus the square root of b squared minus 4ac, all divided by 2a Now your y is equal to, but before that, the value of a is the equation of y squared, which is 1. Alright, so a is equal to 1, b is the coefficient of y, which is minus 10, so b is equal to minus 10. and our c is our constant; c is equal to 1. Okay, all right. Now our y is equal to minus b, that is minus minus 10, which is 10, plus or minus square root of b squared, that is 10 squared, which is 100 minus 4ac, which is 4 times 1 times 1, which will still give us 4, divided by 2a, which is 2 times 1, which will still give us 2. Okay, so if we solve this further, we will have—let me clean this up. side, so let me clean this side; so if we solve this further, we will have that our y is equal to 10 plus or minus the square root of now 100 minus 4 will give us 60—sorry, 100 minus 4 will give us 96. So we have 96 divided by 2, all right? So we have that y is equal to 10 plus or minus now 96. is the same as the square root of 16 multiplied by the square root of 6 divided by 2, all right, so our y is equal to 10 plus or minus the square root of 16, which is 4; we have 4 root 6, okay, divided by 2, so our y is equal to let's factor out 2 here, so 10 divided by 2 gives us 5 plus or minus 4 divided by 2 we get 2 root 6, okay, divided by 2. Now we divide this, and we have that. I want to clean here, but before that I need to write this down because we still need it. So 2 minus 5, 2 divided by 5 raised to the power of x is equal to y. So let me write this in some way. 2 over 5 raised to the power of x is equal to y, all right, so we continue, so we can write that our y is equal to 5 plus or minus 2 root 6, okay. All right, so we call that our y is equal 2 to the power of 2 over 5 to the power of x, so we have 2 divided by 5 to the power of x is equal to 5 plus or minus 2 root 6, okay, so we have 2 over 5 to the power of x now to remove To get the value of x, you need to take the logarithm of both sides of the equation. so when we do that we have log 2 divided by 5 to the power of x is equal to log 5 plus or minus 2 root 6, okay. Now we're going to bring this x down. Why? And that is because when you have log of a to the power of b, this is equal to b multiplied by log a. All right, so because of that we can confidently bring down our x to multiply log 2 over 5, so we have x log 2 over 5 is equal to log 5 plus or minus the square root of 6, okay? Now, for us to get the value of x, we We need to divide both sides of this equation by log 2 over 5, so we have x log 2 over 5. divided by log 2 over 5 okay is equal to log 5 plus or minus square root of 6 divided by log 2 over 5, so we divide this, we have that x is equal to log 5 plus or minus All right, base what? Base 2 over 5? Yes, and that is because when you have log a divided by log b this is the same thing as log a base b, okay, so that simply means that the two values of our x are x is equal to log 5 plus the square root of 6 times 2 over 6 over 5, or x is equal to log 5 minus the square root of 6. 2 over 5, so these are the two values of our x. Thank you so much for watching.