You could have directly assumed the odd / even logic right at the beginning. Since the answer is odd, 8^(y-8) must be 8^(0). Thus directly y = 8. Then apply the remaining logic to derive the value of x.
Fairly simple to do in your head . Most people know 2^10 =1024 so,2^12 =4096 which is 4095 +1 2^12 =2^3^4=8^4 and 1= anything ^0 including 8^0 => 8^4-8^0 =4095 and the rest is easy The driving factor wa"what is the lowesrpower of 2 which is greater than 4095 but divisible by 3 ?' Divisible by three means it is an power of 8,
The solution set is x, y = {12, 8} Working Method. My approach would be to rewrite the right side with a base of 8 - since 4095 = 4096 - 1. 4096 = 2^12 or 8^4; and 1 = 8^0. Therefore; 4095 = 8^4 - 8^0 Accordingly; The entire expression can be presented as: 8^(x - 8) - 8^(y - 8) = 8^4 - 8^0 From this, we can equate exponents and solve the resulting linear equations, thus: x - 8 = 4, and y - 8 = 0 If x - 8 = 4, x = 4 + 8; x = 12 If y - 8 = 0, y = 8 + 0; y = 8 Therefore, the final solution set is x, y = {12, 8} Checking: x = 12, and y = 8 8^(12 - 8) - 8^(8 - 8) = 4095 8^4 - 8^0 = 4095 4096 - 1 = 4095 (Checked).
A unique solution in my style... , let u=8^((x-8)/2) , v=8^((y-8)/2) , u^2=8^(x-8) , v^2=8^(y-8) , u^2-v^2=4095 , (u+v)(u-v)=4095 , etc. , 4095=105*39 , let u+v=105 , u-v=39 , u+v+u-v=105+39 , 105+39=144 , -> 2u=144 , u=72 , u+v=105 , v=105-72 , v=33, u^2=72^2 , u^2=5184 , v^2=33^2 , v^2=1089 , for x , -> 8^(x-8)=5184 , (x-8)*ln8=ln5184 , x-8=ln5184/ln8 , x-8=4.11328 , x=4.11328+8 , x=12.11328 , for y , -> 8^(y-8)=1089 , (y-8)=ln1089/ln8 , y-8=3.36293 , y=11.36293 , test , 8^(x-8)-8^(y-8)=8^(12.11328-8)-8^(11.3693-8) , 8^(12.11328-8)-8^(11.3693-8)=8^(4.11328) - 8^(3.36293) , test -> , 8^4.11328=5184 , 8^3.36293=1089 , 5184-1089=4095 , same , OK , solu. , x=12.11328 , y=11.36293 , /// Large numbers can be tamed by using logarithms... /// , By dividing the problem into products of 4095, there are many, many solutions... etc. , 4095=5*3*7*3*13 , -> 15*273=4095,,, ,
You could have directly assumed the odd / even logic right at the beginning.
Since the answer is odd, 8^(y-8) must be 8^(0). Thus directly y = 8.
Then apply the remaining logic to derive the value of x.
Excellent , again well presented. Thank you.
So nice of you
OBS. 8^(a+y-8)-8^(y-8)=
4095;
8^(y-8)x(8^ a-1)=4095.
Etc.
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Fairly simple to do in your head . Most people know 2^10 =1024 so,2^12 =4096 which is 4095 +1
2^12 =2^3^4=8^4 and 1= anything ^0 including 8^0
=> 8^4-8^0 =4095 and the rest is easy
The driving factor wa"what is the lowesrpower of 2 which is greater than 4095 but divisible by 3 ?' Divisible by three means it is an power of 8,
Greetings. Great job. Absolutely brilliant.
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The solution set is x, y = {12, 8}
Working Method.
My approach would be to rewrite the right side with a base of 8 - since 4095 = 4096 - 1.
4096 = 2^12 or 8^4; and 1 = 8^0.
Therefore; 4095 = 8^4 - 8^0
Accordingly;
The entire expression can be presented as:
8^(x - 8) - 8^(y - 8) = 8^4 - 8^0
From this, we can equate exponents and solve the resulting linear equations, thus:
x - 8 = 4, and y - 8 = 0
If x - 8 = 4, x = 4 + 8; x = 12
If y - 8 = 0, y = 8 + 0; y = 8
Therefore, the final solution set is x, y = {12, 8}
Checking:
x = 12, and y = 8
8^(12 - 8) - 8^(8 - 8) = 4095
8^4 - 8^0 = 4095
4096 - 1 = 4095 (Checked).
A unique solution in my style... , let u=8^((x-8)/2) , v=8^((y-8)/2) , u^2=8^(x-8) , v^2=8^(y-8) , u^2-v^2=4095 , (u+v)(u-v)=4095 ,
etc. , 4095=105*39 , let u+v=105 , u-v=39 , u+v+u-v=105+39 , 105+39=144 , -> 2u=144 , u=72 , u+v=105 , v=105-72 , v=33,
u^2=72^2 , u^2=5184 , v^2=33^2 , v^2=1089 , for x , -> 8^(x-8)=5184 , (x-8)*ln8=ln5184 , x-8=ln5184/ln8 , x-8=4.11328 ,
x=4.11328+8 , x=12.11328 , for y , -> 8^(y-8)=1089 , (y-8)=ln1089/ln8 , y-8=3.36293 , y=11.36293 ,
test , 8^(x-8)-8^(y-8)=8^(12.11328-8)-8^(11.3693-8) , 8^(12.11328-8)-8^(11.3693-8)=8^(4.11328) - 8^(3.36293) ,
test -> , 8^4.11328=5184 , 8^3.36293=1089 , 5184-1089=4095 , same , OK , solu. , x=12.11328 , y=11.36293 ,
/// Large numbers can be tamed by using logarithms... /// ,
By dividing the problem into products of 4095, there are many, many solutions... etc. , 4095=5*3*7*3*13 , -> 15*273=4095,,, ,
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