Solve each equation for the indicated variable. Use logarithms with the appropriate bases. $I=\frac{E}{R}(1-e^{-\frac{Rt}{2}}),\text{ for }t$

Verified step by step guidance

1. Identify the given equation: \(I = \frac{E}{R} \left(1 - e^{-\frac{Rt}{2}}\right)\), and the goal is to solve for the variable \(t\).
2. Isolate the exponential term by first multiplying both sides by \(R\) to get rid of the denominator: \(IR = E \left(1 - e^{-\frac{Rt}{2}}\right)\).
3. Divide both sides by \(E\) to isolate the expression with the exponential: \(\frac{IR}{E} = 1 - e^{-\frac{Rt}{2}}\).
4. Rearrange to isolate the exponential term: \(e^{-\frac{Rt}{2}} = 1 - \frac{IR}{E}\).
5. Take the natural logarithm (ln) of both sides to solve for \(t\): \(-\frac{Rt}{2} = \ln\left(1 - \frac{IR}{E}\right)\), then multiply both sides by \(-\frac{2}{R}\) to isolate \(t\).

Key Concepts

Solving Equations for a Specific Variable

This involves isolating the indicated variable on one side of the equation. It requires algebraic manipulation such as addition, subtraction, multiplication, division, and applying inverse operations to both sides to express the variable explicitly.

Exponential Functions and Their Properties

Exponential functions have variables in the exponent, like e^x. Understanding their behavior and how to manipulate them is essential, especially recognizing that the natural exponential function e^x is the inverse of the natural logarithm.

Logarithms and Their Use in Solving Exponential Equations

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is in the exponent. Applying logarithms with the appropriate base allows you to 'bring down' the exponent and solve for the variable.