Questions: Solve for (x). 16^x+2=32 x=

Solution Steps

To solve the equation \(16^{x+2} = 32\), we can use logarithms to isolate \(x\). First, express both sides of the equation with the same base if possible. Since 16 is \(2^4\) and 32 is \(2^5\), rewrite the equation as \((2^4)^{x+2} = 2^5\). Simplify the left side using the power of a power property, which gives \(2^{4(x+2)} = 2^5\). Since the bases are the same, set the exponents equal to each other and solve for \(x\).

We start with the equation: \[ 16^{x+2} = 32 \] We can express both sides with base \(2\): \[ (2^4)^{x+2} = 2^5 \]

Using the power of a power property, we simplify the left side: \[ 2^{4(x+2)} = 2^5 \] This leads to the equation: \[ 4(x+2) = 5 \]

Now, we solve for \(x\): \[ 4x + 8 = 5 \] Subtracting \(8\) from both sides gives: \[ 4x = -3 \] Dividing by \(4\) results in: \[ x = -\frac{3}{4} \]

The equation also has complex solutions, which are: \[ x = \frac{-\log(8) \pm 2i\pi}{4\log(2)} \] These solutions can be expressed as: \[ x = -\frac{3}{4}, \quad x = \frac{-\log(8)}{4\log(2)} + \frac{2i\pi}{4\log(2)}, \quad x = \frac{-\log(8)}{4\log(2)} - \frac{2i\pi}{4\log(2)} \]

Final Answer