Question

Solve for x 4^x-2=256

Answer 1

We can do this problem using two ways....
Let me show you both methods.

Method 1: "Without using logs"

$4^{x-2}=256\\\\ 4^{x-2}=4^4\ \ \ \ \ |\because 4^4 \ is the same as 256$

Now, since both sides of the equation have same base and they equal each other, then their exponent must be equal to each other. So, we equal their exponents. 

After equaling the exponents, we get,
$x - 2 = 4 \\\\ x=4+2 \\\\ \boxed{x = 6} \Leftarrow This is your answer!!$

Now,
Method 2: "Using Logs"

$4^{x-2}=256 \\\\ ~~~~~~~\Downarrow ~~~~~~ is\ the\ same\ thing\ as\ saying \\\\ log\ _4\ 256 =x-2 \\\\ Now,\ using\ the\ log\ property\ log_AB= \frac{log_cA}{log_cB}, \ we \ get, \\\\ \frac{log_{10}256}{log_{10}4}=x-2 \\\\ 4=x-2 \\\\ 4+2=x\\\\ 6=x \\\\ \boxed{x=6} \Leftarrow This is your answer$



So, either way we get x= 6

Answer 2

$4^{x - 2} = 256 \\ln(4^{x - 2}) = ln(256) \\(x - 2)ln(4) = ln(256) \\\frac{(x - 2)ln(4)}{ln(4)} = \frac{ln(256)}{ln(4)} \\x - 2 = \frac{ln(2^{8})}{ln(2^{2})} \\x - 2 = \frac{8ln(2)}{2ln(2)} \\x - 2 = 4 \\x = 6$