Italy l Very Nice Olympiad Math Exponential Problem l find value of x?

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[@domosautomotive1929]

I'm glad you explained that 1= -1+2, I would have never figured that out

[@borisjeud8464]

X= 2.86...

[@jacobcombs1106]

I don't know why but your way feels like it takes longer to me. Here is how I solved it.
8^x×8-8^x/8=3024
8^x(8-1/8)=3024
8^x(63/8)=3024
8^x=3024×8/63
8^x=384
ln(8^x)=ln(384)
x×ln(8)=ln(384)
x=ln(384)/ln(8)
x=~2.86

[@luiscarloscamargo9941]

Sensacional !!!

[@THyperon]

Everybody has a calculator... so at stage x-1=log48/log8 --> x=1+log48/log8 = 1+1.862 = 2.862 ✅️

[@nucki222]

X = ln 384 / ln 8 ; X = 2,86

[@tonir299]

8^(x+1) - 8^(x-1) = 3024 || *8
8^(x+2) - 8^x = 3024*8
(8^x)*(8^2 - 1) = 3024*8 || ÷63
8^x = 48*8 || factorize
2^(3*x) = 2^7*3 || log
3*x*log2 = 7*log2 + log3 || ÷log2
3*x = 7 + log3 / log2 || ÷3
and change log base to 2
x = (7 + log[2]3) / 3

[@stpat7614]

No fractions in this solution:
8^(x+1) - 8^(x-1) = 3024
8^(x-1+2) - 8^(x-1) = 3024
8^(x-1)*8^2 - 8^(x-1) = 3024
8^(x-1)*64 - 8^(x-1)*1 = 3024
8^(x-1)*(64 - 1) = 3024
8^(x-1)*63 = 3024
8^(x-1)*63/63 = 3024/63
8^(x-1) = 48
log(8^[x-1]) = log(48)
(x-1)*log(8) = log(48)
(x-1)*log(8) = log(6*8)
(x-1)*log(8) = log(6) + log(8)
(x-1)*log(8)/log(8) = (log[6] + log[8]) / log(8)
x - 1 = log(6)/log(8) + log(8)/log(8)
x - 1 = log_8(6) + 1
x - 1 + 1 = log_8(6) + 1 + 1
x = log_8(6) + 2

[@Trog1959]

I'm horrified that this is (supposedly) a math Olympiad question

[@wiseview1444]

trivial

[@NoobVSProlegendary]

Can’t you just make the bases of 8 the same and the power is like 378 so x + 1 x - 1 is equal to 8 to the power of 378

[@RafayKhan-h6q]

😮😮 good