Question

In Exercises, solve for \(x\) or \(t\). $$ e^{-0.5 x}=0.075 $$

Short answer

The solution to the equation is \(x = 5.1806\).

Step-by-step solution

Apply the natural logarithm to both sides

To remove the base \(e\) from the left-hand-side, apply the natural logarithm \(ln\) to both sides of the given equation. The natural logarithm of a number \(b\), denoted \(ln(b)\), is the power to which \(e\) would have to be raised to equal \(b\). So, applying natural log to both sides, the equation becomes: \[ ln(e^{-0.5 x}) = ln(0.075)\]

Use logarithmic identity on the left-hand-side

The natural logarithm of the exponential function is just the function itself. Thereby, \(-0.5x\) is simply left on the left-hand-side: \[-0.5x = ln(0.075)\] The left-hand-side can be rearranged by multiplying by \( -2\), to isolate the variable on the left-hand-side. This results to: \[ x = -2 \times ln(0.075) \].

Evaluate the expression

Finally, calculate the value of the right-hand side using a calculator or any computing tool (like Python, Excel, etc.) that allows the computation of the natural logarithm: \[ x = -2 \times -2.5903.\]