can you solve this wonderful maths equation Problem

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Very nice algebra problem
A nice algebra question
you should try this amazing algebra problem
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Many students failed to solve this tricky algebra problem

[cvdinter]

let log in both side in first step is the common way.

[Ramkabharosa]

7ⁿ⁺⁸ = 8ⁿ⁺⁷, so 7.7ⁿ⁺⁷ = 8ⁿ⁺⁷, and hence 7= 8ⁿ⁺⁷/7ⁿ⁺⁷ = (8/7)ⁿ⁺⁷.
Taking log₁₀ of both sides we get, (n+7).log(8/7) = log(7).
So (n+7) = log(7)/log(8/7). Thus n = log(7)/{log(8)-log(7)} - 7
= (0.84509804 / 0.057991946) - 7 = 7.57267945.

[spondulix]

As noted by Okaro 8 days ago, this result can be somewhat simplified to x = = - [ 7 + log 7 / ( log (7/8) ) ]. And, this can be made even more compact if logarithms to the base 7 are employed. The result is x = - [ 7 + 1 / ( log_7 ( 7/8 ) ) ] or x = 1 / ( log_7 ( 8/7 ) ) - 7 .

[Feanarojp]

Wouldn't it be simpler to start of by multiplying both sides by 8 so you have a common exponent?

[gayesolo]

simplified solution: square both equations and the resulting equation is X-1 + X-2 =1; add like terms and obtain 2x-3=1; 2x=4; divide both sides by 2 then X is equal to 2.

[vpilot]

7.58 (approximately) - computed numerically using lotus 123 spreadsheet.

[satyendrapuri]

It would be much simpler to take log of both sides in the very first step and then solving for x.

[kalyanbhattacharyya]

According to your stated valus of log7 = 0.8951 and log8 =0.9031 though I didn't verify those values with log table, the value of x will be 104.8875 instead of your calculated value 7.571.

[ribo]

Easier:
7^(x+8)=8^(x+7)
exp[(x+8)log7]=exp[(x+7)log8]
(x+8)log7=(x+7)log8
x(log7-log8)=7log8-8log7
etc. (same)

[okaro]

One can simplify that as log 7 / log(8/7) - 7

[bertrandviollet]

If you have (log 5)^x, it's still x.log5?or you must have log(5^x)?

[mandolinic]

YAY! Got it using pretty much the same method as you.

[gelbkehlchen]

Solution:
7^(x+8) = 8^(x+7) ⟹
7^x*7^8 = 8^x*8^7 |/(8^x*7^8) ⟹
(7/8)^x = 8^7/7^8 |ln() ⟹
x*ln(7/8) = ln(8^7/7^8) |/ln(7/8) ⟹
x = ln(8^7/7^8)/ln(7/8) = 7.5726792090532952

sample:
left side: 7^(7.5726792090532952+8) = 14469071986289.74
right side: 8^(7.5726792090532952+7) = 14469071986289.74
left side = right side, all o.k.