Solve Exponential Bcd698

1. **State the problem:** We need to find the value of $t$ that satisfies the equation $$5(2^t) = 4.$$\n\n2. **Isolate the exponential term:** Divide both sides of the equation by 5 to isolate $2^t$.\n$$\cancel{5}(2^t) = \frac{4}{\cancel{5}} \implies 2^t = \frac{4}{5}.$$\n\n3. **Apply logarithms:** To solve for $t$, take the logarithm base 2 of both sides. Using the property $\log_b(a^c) = c \log_b(a)$, we get:\n$$t = \log_2\left(\frac{4}{5}\right).$$\n\n4. **Convert to natural logarithm (optional):** Using change of base formula,\n$$t = \frac{\ln\left(\frac{4}{5}\right)}{\ln(2)}.$$\n\n5. **Calculate the value:**\nCalculate numerator: $\ln\left(\frac{4}{5}\right) = \ln(0.8) \approx -0.2231$.\nCalculate denominator: $\ln(2) \approx 0.6931$.\n\n6. **Divide:**\n$$t \approx \frac{-0.2231}{0.6931} \approx -0.322.$$\n\n**Final answer:**\n$$t = -0.322.$$