Questions: Find the solution for t in the equation 5(2^∘)=4. t=-0.322 t=0.861 t=0.322 t=-0.861

Solution Steps

To solve the equation \(5 \cdot 2^t = 4\) for \(t\), we need to isolate \(t\). This can be done by first dividing both sides by 5, and then taking the logarithm of both sides to solve for \(t\).

Given the equation: \[ 5 \cdot 2^t = 4 \] Divide both sides by 5: \[ 2^t = \frac{4}{5} \]

Take the natural logarithm (ln) of both sides: \[ \ln(2^t) = \ln\left(\frac{4}{5}\right) \]

Using the property of logarithms \(\ln(a^b) = b \cdot \ln(a)\): \[ t \cdot \ln(2) = \ln\left(\frac{4}{5}\right) \]

Isolate \( t \) by dividing both sides by \(\ln(2)\): \[ t = \frac{\ln\left(\frac{4}{5}\right)}{\ln(2)} \]

Using the values: \[ \ln\left(\frac{4}{5}\right) \approx -0.2231 \] \[ \ln(2) \approx 0.6931 \] \[ t = \frac{-0.2231}{0.6931} \approx -0.322 \]

Final Answer