Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. r = p - k ln t, for t

Verified step by step guidance

1. Start with the given equation: $r=p-k\ln t$.
2. Isolate the logarithmic term by subtracting $p$ from both sides: $r-p=-k\ln t$.
3. Divide both sides by $-k$ to solve for $\ln t$: $\ln t=\frac{p-r}{k}$.
4. Exponentiate both sides to eliminate the natural logarithm: $t=e^{\frac{p-r}{k}}$.
5. The expression $t=e^{\frac{p-r}{k}}$ is the solution for $t$.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve equations where the variable is an exponent. They allow us to express relationships in multiplicative terms as additive ones, making it easier to isolate variables. Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating equations involving logarithms.

Isolating Variables

Isolating a variable involves rearranging an equation to solve for that specific variable. This process often requires the use of algebraic operations such as addition, subtraction, multiplication, and division. In the context of the given equation, isolating 't' means expressing it in terms of 'r', 'p', and 'k', which may involve applying logarithmic properties to simplify the equation.

Natural Logarithm

The natural logarithm, denoted as 'ln', is a logarithm with base 'e', where 'e' is approximately equal to 2.71828. It is commonly used in calculus and mathematical modeling due to its unique properties, particularly in growth and decay problems. In the equation provided, using the natural logarithm is crucial for solving for 't' when 'ln t' is present, as it allows for the manipulation of the equation to isolate 't' effectively.