Question

4^x - 10*4^-x = 3

Answer 1

okay, so we have:
\[4^x - (10) 4^{-x}=3\]
Now, using some exponential properties, we will re-write the equation, more friendly for operating:
\[4^x -\frac{ 10 }{ 4^{x} }=3\]
Okay, that's as far as I can simplify it, but I have no properties that allow me to associate sum or sustraction of exponential equalities or exponents in general.
So, in the way of math, when we encounter something like this, we do a "change of variable".
This means that I will assingn the convinient variable value, in order to operate this, in an equal way, and simplify it sufficiently to find the value we are looking for.
And since I cannot simplify the expression any further I'll apply it:
Let:
\[G=4^x\]
So, now, where I see "4^x" I'll write "G" instead.
\[G-\frac{ 10 }{ G }=3\]
Now this is better, it looks like the single variable equations we did in our first highscool math classes, let's solve for "G":
\[\frac{ G^2-10 }{ G }=3\]
All I did was apply common denominator and now, I'll multiply both sides by "G":
\[G^2-10=3G\]
Then:
\[G^2-3G-10=0\]
and this is a quadratic equation we can solve with the general formula, It'll spit out two values of "G" , these values will be:
\[G=-2\]
\[G=5\]
Now, let's replace the original value of G in both:
\[4^x=2 => x=1/2\]
\[4^x = 5\]
\[Log4^x=Log 5\]
\[x=\frac{ Log5 }{ Log4 }\]
So x then has two possible values:
\[x _{1}= 1/2\]
\[x _{2}=\frac{ \log_{} 5 }{ \log_{} 4 }\]

Answer 2

there is an error in the solutions

x = 1/2 can't be a solution

the factor is

\[4^x + 2 = 0\]

so

\[4^x = -2\]

this equation has no solution...

you can't find a power that gives a negative value....

the 2nd solution is correct

x = log(5)/log(4)

Answer 3

^ Watch out for signs hehehehe.
Thanks a lot, Campbell