Solving 16•4^{-x}=4^x-6 Equation

[Petrus]

Hello MHB,
I am stuck on this equation and don't know what to do, If i take ln it does not work, any advice?
\(\displaystyle 16•4^{-x}=4^x-6\)

Regards,
\(\displaystyle |\pi\rangle\)

 

[Nono713]

Try multiplying both sides by $4^x$. You get:
$$16 = (4^x)^2 - 6 \cdot 4^x$$
Now substitute $u = 4^x$. What do you see? :)

 

[chisigma]

Hello MHB,
I am stuck on this equation and don't know what to do, If i take ln it does not work, any advice?
\(\displaystyle 16•4^{-x}=4^x-6\)

Regards,
\(\displaystyle |\pi\rangle\)

Set $4^{x}=y$, then solve for y and finally find $x = \frac{\ln y}{\ln 4}$ ...

Kind regards

$\chi$ $\sigma$

 

[Petrus]

Hello,
Thanks for the fast respond and help from you both!:) i succed to solve it with correct answer!:)
Regards,
\(\displaystyle |\pi\rangle\)

 

[CellCoree]

Hello |\pi\rangle,

To solve this equation, we can use the properties of logarithms to rewrite it in a more manageable form. First, let's divide both sides by 4^x to get:

16•4^{-x}/4^x = (4^x-6)/4^x

Next, we can simplify the left side using the power rule of logarithms:

16•(4^x)^{-1} = (4^x-6)/4^x

Now, let's use the fact that (a^b)^c = a^{bc} to rewrite the left side as:

16•(4^{x\cdot -1}) = (4^x-6)/4^x

Using the fact that a^{-1} = 1/a, we can simplify the left side to get:

16•(1/4^x) = (4^x-6)/4^x

Now, we can use the fact that (a/b)^c = a^c/b^c to rewrite the left side as:

16/4^x = (4^x-6)/4^x

Finally, we can simplify the right side by dividing both the numerator and denominator by 4^x to get:

16/4^x = (4^x/4^x)-(6/4^x)

Using the fact that a^0 = 1, we can simplify the left side to get:

16/4^x = 1-6/4^x

Now, we have a simpler equation to solve:

16/4^x = 1-6/4^x

To solve for x, we can multiply both sides by 4^x to get:

16 = 4^x - 6

Adding 6 to both sides, we get:

22 = 4^x

Taking the logarithm of both sides, we get:

ln(22) = ln(4^x)

Using the fact that ln(a^b) = b•ln(a), we can rewrite the right side as:

ln(22) = x•ln(4)

Finally, we can solve for x by dividing both sides by ln(4):

x = ln(22)/ln(4)

Using a calculator, we can approximate this value to be around 1.415. Therefore, the solution to the equation is x = 1.