I have this equation:  $3^x + 4^x = 5^x$, according to , $x &lt;= 2$, so the answer is $2$.  But how can I get to the result through an elementary algebraic procedure?  I have tried many things, but I have not come up with any concrete results, arriving only at:  First try, $log(3^x+4^x) = 3log5$  Second try, $(\frac{3}{5})^x + (\frac{4}{5})^x = 1$  I have also tried to search several programs, such as symbolab, mathway, etc. But, they have not been able to solve it, the only program that has given me an answer is wolframAlpha, but I can not visualize it step by step, thanks in advance.
I would like to prove that the equation $ 3^x+4^x=5^x $ has only one real solution ($x=2$)  I tried to study the function $ f(x)=5^x-4^x-3^x $ (in order to use the intermediate value theorem) but I am not able to find the sign of $ f&#39;(x)= \ln(5)\times5^x-\ln(4)\times4^x-\ln(3)\times3^x $ and I can't see any other method to solve this exercise...
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It's probably pretty trivial to implement, and nice to have! I have this problem.  I need to do some administrative tasks on a MS SQL Database using SA account.   OS authentication is not set. SQL Server Manager Manager Studio is not installed. I have no other admin database account. I do have access to a Windows admin account The only tools installed on the database server are as shown:     How can I activate OS authentication so I can log into the database an reset the SA password ? Tachyons and Photons Is there a particle called a "tachyon" that can travel faster than light?  If so, would Einstein's relativity be wrong?  According to Einstein no particle can travel faster than light. If column matches another file, print every line with match (awk/grep) I’m taking two input files, one with certain ID numbers, and another with a large list of ID numbers and additional columns. The latter file contains multiple lines for each ID number and I need to extract all lines that match an ID from the first file. Those lines then must be printed in a new file.  Edit 1: Replaced sample files with excerpts from actual  Edit 2: Removed extra spaces that were in excerpt, but not actual file. Files likely need to be sanitized in some way, but how is unclear.  file1:  AT1G56430 AT3G55190 AT3G22880   file2:  AT1G01010|GO:0043090|RCA AT1G56430|GO:0010233|IGI  AT1G56430|GO:0009555|IGI  AT1G56430|GO:0030418|IGI   expected output  AT1G56430|GO:0010233|IGI  AT1G56430|GO:0009555|IGI  AT1G56430|GO:0030418|IGI   [[  I have tried:  awk -F'|' 'NR==FNR{c[$1$2]++;next};c[$1$2] &gt; 0' file1 file2 &gt; output.txt   and:  grep -Ff file2 file1 &gt; output.txt   I’m aware that there are many somewhat similar questions posted in these forums and others. However, these don’t mention how the output is handled… nor do they mention duplicates. 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