If $5+20 * 2^{2-3 x}=10 * 2^{-2 x}+5$, what is the value of $x$?
The given equation is \(5 + 20 \cdot 2^{2-3x} = 10 \cdot 2^{-2x} + 5\). We aim to find the real value of \(x\) that satisfies this equality. The equation involves exponential expressions with base 2, which allows us to simplify the expression by manipulating the exponents and constants. We can start by simplifying the constant terms on both sides of the equation.
We subtract 5 from both sides of the equation:
\[5 + 20 \cdot 2^{2-3x} - 5 = 10 \cdot 2^{-2x} + 5 - 5\]
This simplifies to:
\[20 \cdot 2^{2-3x} = 10 \cdot 2^{-2x}\]
Next, we divide both sides of the equation by 10 to isolate the exponential terms:
\[\frac{20 \cdot 2^{2-3x}}{10} = \frac{10 \cdot 2^{-2x}}{10}\]
\[2 \cdot 2^{2-3x} = 2^{-2x}\]
On the left side, we use the property of exponents that \(a^m \cdot a^n = a^{m+n}\). Since \(2\) can be written as \(2^1\), the left side becomes:
\[2^1 \cdot 2^{2-3x} = 2^{1 + 2 - 3x} = 2^{3-3x}\]
Now the equation is:
\[2^{3-3x} = 2^{-2x}\]
Since the bases are identical, we can set the exponents equal to each other:
\[3 - 3x = -2x\]
Now we solve for \(x\):
\[3 = 3x - 2x\]
\[3 = x\]
Therefore, the value of \(x\) that satisfies the equation is 3. Comparing this to the given options, the correct choice is 3.
Exponential equations appear frequently in science and finance. For instance, if a radioactive substance decays or an investment grows, the population or amount often follows an exponential model. By solving for the variable in the exponent, we can determine the exact time it takes for a substance to reach a certain level or for an investment to grow to a specific amount. Mastering these techniques is essential for predicting trends in various real-world scenarios.