logarithms

**prasum** · October 28th 2010, 09:30 PM

find x if 3^(log base (a) x)+3*x^(log base(a) 3)=2

please help

**prasum** · October 29th 2010, 02:22 AM

oh i got the answer we know by property that the two logs in question would be equal

then we can solve it

**Soroban** · October 29th 2010, 05:06 AM

Hello, prasum!

I don't understand your reasoning.
Will you tell us your answer?

I think I solved it . . . and got an ugly answer.

$\text{Find }x\text{ if: }\;3^{\log_ax} +3\cdot x^{\log_a3} \:=\:2$

$\text{Let: }\:3^{\log_ax} \:=\;P$

$\text{Take logs, base 3: }\;\log_3\!\left(3^{\log_ax}\right) \;=\;\log_3(P)$

. . . . . . . . . . . $\log_a(x)\cdot\underbrace{\log_3(3)}_{\text{This is 1}} \;=\;\log_3(P)$
. . . . . . . . . . . . . . . . . $\log_a(x) \;=\;\dfrac{\log_a(P)}{\log_a(3)}$

. . . . . . . . . . . . . . . . . $\log_a(P) \;=\;\log_a(x)\cdot\log_a(3)$

. . . . . . . . . . . . . . . . . . . . . $P \;=\;a^{\log_a(x)\cdot\log_a(3)}$

. . . . . . . . . . . . . . . . . . . . . $P \;=\;\left(a^{\log_a(x)}\right)^{\log_a(3)}$

. . . . . . . . . . . . . . . . . . . . . $P \;=\;x^{\log_a(3)}$

Hence: . $3^{\log_a(x)} \;=\;x^{\log_a(3)}$

The equation becomes: . $x^{\log_a(3)} + 3\cdot x^{\log_a(3)} \;=\;2$

. . . . . . . . . . . . . . . . . . . . . . . . $4\cdot x^{\log_a(3)} \;=\;2$

. . . . . . . . . . . . . . . . . . . . . . . . . . $x^{\log_a(3)} \;=\;\dfrac{1}{2}$

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $x \;=\;\left(\dfrac{1}{2}\right)^{\frac{1}{\log_a(3) }}$

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $x \;=\;\left(\dfrac{1}{2}\right)^{\log_3(a)}$

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