Question

Solve: log5x3 - logx2 = 2

x =

Answer 1

Interpret the equation as log_5(x^3) − log_x(2) = 2. (So x>0, x≠1.)

Write logs in terms of natural log: 3 log_5 x − log_x 2 = 2
=> 3 (ln x / ln 5) − (ln 2 / ln x) = 2.

Let t = ln x. Multiply through by ln 5 · t:
3 t^2 − 2(ln 5) t − (ln 2)(ln 5) = 0.

Solve the quadratic for t:
t = [2 ln 5 ± sqrt(4(ln 5)^2 + 12 ln 2 · ln 5)] / 6
= [ln 5 ± sqrt(ln 5 · ln 40)] / 3.

Thus
x = exp(t) = exp( (ln 5 ± sqrt(ln 5 · ln 40))/3 ).

Numerically,
x ≈ 3.8530 or x ≈ 0.7589.

Both are valid (positive and ≠1).