Question 271636
<pre><font size = 4 color = "indigo"><b>
{{{3^ (x* sqrt(2)) + 4*3^(x*sqrt(8)) -3 =0}}}

{{{3^ (x* sqrt(2)) + 4*3^(x*sqrt(4*2)) -3 =0}}}

{{{3^ (x* sqrt(2)) + 4*3^(x*2sqrt(2)) -3 =0}}}

{{{3^ (x* sqrt(2)) + 4*(3^(x*sqrt(2)))^2 -3 =0}}}

{{{4*(3^(x*sqrt(2)))^2 + 3^ (x* sqrt(2)) -3 =0}}}

{{{4*(3^(x*sqrt(2)))^2 + (3^ (x* sqrt(2))) -3 =0}}}

Let some letter other than {{{x}}}, say {{{t}}},

equal to {{{(3^ (x* sqrt(2)))}}}. 

Then we have {{{t=(3^ (x* sqrt(2)))}}}.  Substituting,

{{{4t^2 + t -3 =0}}}

Factoring:

{{{4t^2 + t -3 =0}}}

{{{(4t-3)(t+1)=0}}}

Use the zero factor principle.

Set the first factor = 0

{{{4t-3=0}}}
{{{4t=3}}}
{{{t=3/4}}}

Setting second factor = 0

{{{t+1=0}}}
{{{t=-1}}}

Using first solution for {{{t}}} in

{{{t=(3^ (x* sqrt(2)))}}},

{{{3/4=(3^ (x* sqrt(2)))}}}

Take logs of both sides

{{{log((3/4))=log((3^ (x* sqrt(2))))}}} 

Use rules of logarithms:

{{{log((3))-log((4))=x* sqrt(2)*log((3))}}}

Divide both sides by {{{sqrt(2)*log((3))}}}

{{{(log((3))-log((4)))/(sqrt(2)*log((3)))=x}}} 

Get a calculator and punch out the left side:

{{{-.1851626332}}}

That's one solution.  We must also consider 

the solution for t {{{t=-1}}} to see if that 

also yields a solution for x:

Using this solution for {{{t}}} in

{{{t=(3^ (x* sqrt(2)))}}}.

{{{-1=(3^ (x* sqrt(2)))}}}.

No power of 3 can ever be negative, so we discard
the solution for {{{t}}}, {{{t=-1}}} as extraneous.

The only solution is  

{{{-.1851626332}}}, rounded to tenths is t=-.2

Edwin</pre>