Polynomial with variable in the exponent.

Question

How do I solve a polynomial with the variable in the exponent? For the following question how would I find n? I tried to log both sides but that failed. Note: Both exponents should be negative.

Answer 1

Use the following substitution:$$x=1.1^n$$$$4.7992=[(1.1)^n]+2[(1.1)^n]^{-2}$$$$4.7992=x+2x^{-2}$$$$4.7992x^2=x^3+2$$$$0=x^3-4.7992x^2+2$$

Use approximations from here, e.g. linear approximation.