Solve for x:
$16^{2x+3} = 2^{3x+2}$
We are given the exponential equation \(16^{2x+3} = 2^{3x+2}\). To solve for \(x,\) we should first express both sides of the equation with a common base. Since \(16\) is a power of \(2\), we can write \(16 = 2^4\). By substituting this into the equation, we can convert both sides into expressions with the base \(2\).
Replacing \(16\) with \(2^4\), the left side becomes \((2^4)^{2x+3}\). According to the power of a power rule in exponents, \((a^m)^n = a^{m \cdot n}\), so we have:
\[(2^4)^{2x+3} = 2^{4(2x+3)} = 2^{8x+12}.\]
Now the equation is:
\[2^{8x+12} = 2^{3x+2}.\]
Since the bases on both sides are now identical (\(2\)), their exponents must be equal for the equation to hold true. Therefore, we can set the exponents equal to each other:
\[8x + 12 = 3x + 2.\]
Now we solve for \(x\) by isolating the variable on one side. Subtract \(3x\) from both sides:
\[8x - 3x + 12 = 2\]
\[5x + 12 = 2\]
Next, subtract \(12\) from both sides:
\[5x = 2 - 12\]
\[5x = -10\]
Finally, divide by \(5\):
\[x = \frac{-10}{5} = -2.\]
To verify our result, substitute \(x = -2\) back into the original equation:
Left side: \(16^{2(-2)+3} = 16^{-4+3} = 16^{-1} = \frac{1}{16}\).
Right side: \(2^{3(-2)+2} = 2^{-6+2} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16}\).
Since both sides equal \(\frac{1}{16}\), our solution \(x = -2\) is correct. Thus, the final value is \(\boxed{-2}\).
Exponential equations are essential for understanding growth and decay, such as calculating the time it takes for a savings account with compound interest to reach a certain value, or modeling population growth over time. For example, if a company's profit doubles every year, you can model their growth with an exponential equation to determine when they will reach a specific target, much like finding the value of \(x\) in the equation above. Mastering these algebraic techniques provides a foundation for solving these real-world problems.