x=log((1+√5)/2)/log(7) is the only real solution: 49^x-7^x-1^x=0 is equivalent to (7^x)^2-7^x-1=0. This is a quadratic equation in 7^x. Use the formula for quadratic equations to find 7^x=(1±√5)/2. From this we have x=log((1±√5)/2)/log(7), but only the positive sign gives a real solution.
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x=log((1+√5)/2)/log(7) is the only real solution:
49^x-7^x-1^x=0 is equivalent to (7^x)^2-7^x-1=0. This is a quadratic equation in 7^x. Use the formula for quadratic equations to find 7^x=(1±√5)/2. From this we have x=log((1±√5)/2)/log(7), but only the positive sign gives a real solution.
the solution is x=log_7 (1+√5)/2