The time t in years for an amount of money invested at an...

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

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

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Key Concepts

Logarithmic Functions

Logarithmic functions serve as the inverse of exponential functions and are essential for solving equations where the unknown variable is an exponent. In the context of exponential growth and compound interest, logarithms are used to isolate time variables and calculate periods such as doubling time.

Doubling Time

Doubling time refers to the period required for a quantity, growing exponentially, to become twice its initial value. It is determined by solving an exponential growth equation, often through the use of logarithms, and is a practical measure in assessing investment growth and other similar contexts.

Exponential Growth

Exponential growth describes the process by which a quantity increases at a rate proportional to its current value, leading to rapid escalation over time. This concept is widely applicable in various fields such as population dynamics, finance, and natural phenomena, where growth compounds over equal intervals.

Compound Interest

Compound interest is the method of calculating interest where accrued interest is added to the principal, resulting in future interest being earned on both the initial amount and the accumulated interest. This cumulative effect is at the heart of many financial growth models and investment strategies.

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