Solving Exponential Equations: Unveiling the Logarithmic Solution and Approximating the Value of t

First, let's manipulate the equation to get t by itself. The equation given is -5e^(10t) = -30. This can be rewritten as e^(10t) = 6 (since -30/-5 = 6). To isolate t, we will next take the natural logarithm (the logarithm base e) of both sides. Therefore, you get 10t = ln(6). From here, divide the equation by 10 to solve for t, thus getting t = ln(6)/10. Note: We can't express the solution in base -e since the base of a logarithm is always a positive real number other than 1. Now, regarding the approximation of the value of t, you would plug ln(6)/10 into a calculator to get the decimal representation of 't'. Doing this, you would find that t ≈ 0.179. After rounding to the nearest thousandth, this gives you t = 0.179.