Solving Exponential Equations: Unveiling the Enigmatic "t" in a Logarithmic World

We start by dividing both sides of the equation (-5e^10t = -30) by -5: e^10t = 6 Next, we take the natural logarithm, ln( ), of both sides. Remember that the logarithm base e is the natural logarithm, so we write: 10t * ln(e) = ln(6) Since ln(e) = 1, it simplifies to: 10t = ln(6) Now, solve for 't' by dividing both sides by 10: t = ln(6) / 10 So t is expressed as a logarithm in base e. To find the approximate value of t, substitute ln(6) with its value (approximately ~1.792), then divide by 10 to obtain the value of 't': t = 1.792 / 10 = 0.1792 Rounding your answer to the nearest thousandth, t = 0.179.