Question

Solve the equation log3 (x + 17) - 2 = log3 2x

Answer 1

\[ log_3 (x + 17) - 2 = log_3 2x\]\[ log_3 (x + 17) - 2log_33 = log_3 2x\]\[ log_3 (x + 17) - log_33^2 = log_3 2x\]\[ log_3 (\frac{x + 17 }{3^2} )= log_3 2x\]\[(\frac{x + 17 }{3^2} )= 2x\]I think you can do it from here

Answer 2

Things to note:
(i) \(log_33 = 1\)
(ii) \(n\log a = \log a^n\)
(iii) \(\log a - \log b = \log \frac{a}{b}\)
(iv) when you see \(log_n (x) = log_n (y)\), you can get x = y, where x and y are some polynomials (or monomials, it depends)

Answer 3

For (i), to generalise it, you may say \(log_nn=1\)

Answer 4

Note that it is for positive numbers. I am not sure of the negative one.