The time t in years for an amount increasing at a rate of r...

The time $t$ in years for an amount increasing at a rate of $r$ (in decimal form) to double is given by $$t(r)=\frac{\ln 2}{\ln (1+r)}$$ This is called doubling time. Find the doubling time to the nearest tenth for an investment at each interest rate. (a) $2 \% \text { (or } 0.02)$ (b) $5 \% \text { (or } 0.05)$ (c) $8 \% \text { (or } 0.08)$

The time $t$ in years for an amount increasing at a rate of $r$ (in decimal form) to double is given by
$$t(r)=\frac{\ln 2}{\ln (1+r)}$$
This is called doubling time. Find the doubling time to the nearest tenth for an investment at each interest rate.
(a) $2 \% \text { (or } 0.02)$
(b) $5 \% \text { (or } 0.05)$
(c) $8 \% \text { (or } 0.08)$

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Key Concepts

Logarithms

Logarithms serve as the inverse of exponential functions and are used to solve equations where the unknown appears in the exponent. They are essential in deriving formulas such as the doubling time formula, enabling the isolation and calculation of the time variable in exponential scenarios.

Exponential Growth

Exponential growth describes processes in which quantities increase at a rate proportional to their current value, leading to accelerating growth over time. This concept is foundational in understanding how values such as investments or populations can double under consistent compounding conditions.

Doubling Time

Doubling time is the period required for a quantity growing exponentially to become twice its initial size. It is often calculated using logarithms, providing insight into the speed at which compounding processes accumulate over time.

Compound Interest

Compound interest is the practice of earning interest on both the initial principal and the accumulated interest from prior periods. This concept is central in finance, illustrating how reinvesting earnings accelerates the growth of investments over time.

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