1. ## Exponents

if 4^x-4^(x-1)=24, What is the value of (2x)^x

I think that x=2.5 I used my calculator and not quite sure. I also am not quite sure how to do this by hand because on my test I cannot use the calculator. I can figure out by starting at zero and going up bu 1 its .75, 3, 12, 48 so so 24 is 2 times 12 so it is half between 2 and 3 thus 2.5. There has to be an easier way. Please help me

2. Originally Posted by IDontunderstand
if 4^x-4^(x-1)=24, What is the value of (2x)^x

I think that x=2.5 I used my calculator and not quite sure. I also am not quite sure how to do this by hand because on my test I cannot use the calculator. I can figure out by starting at zero and going up bu 1 its .75, 3, 12, 48 so so 24 is 2 times 12 so it is half between 2 and 3 thus 2.5. There has to be an easier way. Please help me
$4^x-4^{(x-1)}=24 \quad \Rightarrow 4^x- \{ 4^x \cdot 4^{-1} \}=24$
$\Rightarrow 4^x- \frac{4^x}{4}=24 \quad \Rightarrow 4^x(1- \frac{1}{4})=24 \quad \Rightarrow 4^x \cdot \frac{3}{4}=24$
$\Rightarrow 4^x = \frac{24 \times 4}{3} \quad \Rightarrow (2^2)^x=32$
$\Rightarrow 2^{2x}=2^5 \quad \Rightarrow 2x=5 \quad \Rightarrow\boxed {x=\frac{5}{2}}$

$\therefore (2x)^x =(2 \cdot \frac{5}{2})^{\frac{5}{2}}= 5^{\frac{5}{2}}=25 \sqrt {5}$

3. 4^x - 4^(x-1) = 24
4^x - 4^x(4^-1) = 24
4^x(1 - 4^-1) = 24
4^x(1 - 1/4) = 24
4^x(3/4) = 24
4^x = 24 / (3/4)
4^x = 32
x = log(32) / log(4) = 2.5