Question

If log3 x + logx 3= 2.5 find the value of x

Answer 1

log₃x = 1/log_x3

so, using base 3, we have
logx + 1/logx = 5/2
2(logx)^2 + 2 = 5logx
2(logx)^2 - 5logx + 2 = 0
(2logx-1)(logx-2) = 0
logx = 1/2 or 2
x = √3 or 9

Or, using the change-of-base formula, we have, using any base,
logx/log3 + log3/logx = 5/2
2log²x - 5log3 logx + 2log²3 = 0
(2logx-1)(logx-2) = 0
logx = 1/2 or 2
since this is true for any base, it is true for base 3, so we have
x = √3 or 9