The equation $3^{x-1}+5^{x-1}=34$ has

It is quite clear that x=3 equation we now show that there is no other solution. Note that y = 3^x–1 and y = 5^x–1 are both increasing functions of x (exponential functions with base greater than 1). Therefore
their sum y = 3^x–1 + 5^x–1 is also an increasing

function of x. This means for x < 3, 3^x–1 + 5^x–1 < 34 and for x > 3, 3^x–1 + 5^x–1 > 34. Thus, the equation has no other
solution. (Fig. 2.16)