Before viewing: Set Y = 5^x. Then 5^3x = (5^x)^3 = y^3. Then you have y^3 + y = 10 Subtract 10 from both sides> Y^3 + y -10 = 0 Now we have to get into synthetic division. You can look up how to do this online (chilimath has an easily-digestible explanation). Using c=2, we now factor this into: (Y-2)(y^2+2y+5)=0 One solution, then, is y=2 (AKA 5^x=2) . As for the second, we check the discriminant quickly to see if we should use the quadratic formula. Finding that the discriminant is negative, we know our other two roots will be complex (in fact they will be -1±2i). In general, unless your teacher/professor specifically instructs you to work with the complex roots, we discard those. So then, we have 5^x = 2. Taking the natural log of both sides: x ln 5 = ln 2. And finally divide both sides by ln 5... x = (ln 2)/(ln 5) (You'll do something similar with the complex roots, if you're required to do so. ln(-1±2i)/ln(5) should be approximately 1/2 ± 1.2641)
