Solve for \( x: \quad 4^{x+1}=45 \)

To solve the equation \( 4^{x+1} = 45 \), we can start by rewriting the left side using properties of exponents. Since \( 4 \) is the same as \( 2^2 \), we can express the equation as: \[ (2^2)^{x+1} = 45 \] This simplifies to: \[ 2^{2(x+1)} = 45 \] Next, we can take the logarithm of both sides to solve for \( x \): \[ \log(2^{2(x+1)}) = \log(45) \] Applying the logarithmic identity \( \log(a^b) = b \cdot \log(a) \), we have: \[ 2(x+1)\log(2) = \log(45) \] Now, divide both sides by \( 2\log(2) \): \[ x + 1 = \frac{\log(45)}{2\log(2)} \] By subtracting \( 1 \) from both sides, we can isolate \( x \): \[ x = \frac{\log(45)}{2\log(2)} - 1 \] Now you can calculate the values of \( \log(45) \) and \( \log(2) \) using a calculator to find the numerical solution for \( x \).