Questions: (13/5)^x = 4^(x+1)

Solution Steps

To solve the equation \(\left(\frac{13}{5}\right)^{x}=4^{x+1}\), we can take the natural logarithm of both sides to simplify the exponents. This will allow us to solve for \(x\) using algebraic manipulation.

Given the equation: \[ \left(\frac{13}{5}\right)^{x} = 4^{x+1} \] we take the natural logarithm of both sides: \[ \ln\left(\left(\frac{13}{5}\right)^{x}\right) = \ln\left(4^{x+1}\right) \]

Using the property \(\ln(a^b) = b \ln(a)\), we get: \[ x \ln\left(\frac{13}{5}\right) = (x + 1) \ln(4) \]

Distribute \(\ln(4)\) on the right-hand side: \[ x \ln\left(\frac{13}{5}\right) = x \ln(4) + \ln(4) \] Rearrange to isolate \(x\): \[ x \ln\left(\frac{13}{5}\right) - x \ln(4) = \ln(4) \] Factor out \(x\): \[ x \left(\ln\left(\frac{13}{5}\right) - \ln(4)\right) = \ln(4) \]

Divide both sides by \(\ln\left(\frac{13}{5}\right) - \ln(4)\): \[ x = \frac{\ln(4)}{\ln\left(\frac{13}{5}\right) - \ln(4)} \]

Substitute the values: \[ \ln\left(\frac{13}{5}\right) \approx 0.9555 \] \[ \ln(4) \approx 1.3863 \] \[ x = \frac{1.3863}{0.9555 - 1.3863} \approx -3.2181 \]

Final Answer