Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. y = A + B(1 - e^(-Cx)), for x

Verified step by step guidance

1. Start by isolating the exponential term: subtract A from both sides to get y - A = B(1 - e^{-Cx}).
2. Divide both sides by B to isolate the exponential expression: \( \frac{y - A}{B} = 1 - e^{-Cx} \).
3. Rearrange the equation to isolate the exponential term: \( e^{-Cx} = 1 - \frac{y - A}{B} \).
4. Take the natural logarithm of both sides to solve for \(-Cx\): \( -Cx = \ln\left(1 - \frac{y - A}{B}\right) \).
5. Finally, solve for \(x\) by dividing both sides by \(-C\): \( x = -\frac{1}{C} \ln\left(1 - \frac{y - A}{B}\right) \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the equation provided, e^(-Cx) represents an exponential decay function, where 'e' is the base of natural logarithms. Understanding how these functions behave is crucial for manipulating and solving equations involving them.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations. When dealing with equations like y = A + B(1 - e^(-Cx)), logarithms help isolate the variable x by transforming the exponential part into a linear form. Familiarity with properties of logarithms, such as the change of base formula, is essential for solving such equations.

Isolating Variables

Isolating a variable involves rearranging an equation to solve for that specific variable. In the context of the given equation, this means manipulating the equation step-by-step to express x in terms of the other variables. Mastery of algebraic techniques, such as addition, subtraction, multiplication, and division, is necessary to effectively isolate the desired variable.