logarithms equations, different basessolve equations: log x ⁡ 10 + 2 log 10 x ⁡...

Rapsinincke

Rapsinincke

Answered

2022-07-08

logarithms equations, different bases
solve equations: log x 10 + 2 log 10 x 10 3 log 100 x 10 = 0 so I tried to use log a b = 1 log b a but it didn't work for me.

Answer & Explanation

Oliver Shepherd

Oliver Shepherd

Expert

2022-07-09Added 24 answers

log 10 x 10 = 1 log 10 ( 10 x ) = 1 1 + log 10 x = 1 1 + u
and similarly
log 100 x 10 = 1 log 10 ( 100 x ) = 1 2 + log 10 x = 1 2 + u .
So you have
1 u + 2 1 + u 3 2 + u = 0.
If you multiply both sides by u ( 1 + u ) ( 2 + u ), you get
( 1 + u ) ( 2 + u ) + 2 u ( 2 + u ) 3 u ( 1 + u ) = 0.
Multiply that out, then collect like terms, then you have a quadratic equation.

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