If an investment is growing continuously for t years, its...

If an investment is growing continuously for $t$ years, its anmal growth rate $r$ is given by the following formula, where $P$ is the current value and $P_{0}$ is the amount originally invested. $$r=\frac{1}{t} \ln \frac{P}{P_{0}}$$ Suppose you graph the function $f(x)=\ln x$ on a coordinate grid with a unit distance of 1 centimeter on the $x$ - and $y$ -axes. How far out must you go on the $x$ -axis so that $f(x)=12 ?$ Give your result to the nearest mile.

If an investment is growing continuously for $t$ years, its anmal growth rate $r$ is given by the following formula, where $P$ is the current value and $P_{0}$ is the amount originally invested.
$$r=\frac{1}{t} \ln \frac{P}{P_{0}}$$
Suppose you graph the function $f(x)=\ln x$ on a coordinate grid with a unit distance of 1 centimeter on the $x$ - and $y$ -axes. How far out must you go on the $x$ -axis so that $f(x)=12 ?$ Give your result to the nearest mile.

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Key Concepts

Natural Logarithm Function

The natural logarithm, denoted ln(x), is the inverse of the exponential function with base e. It calculates the power to which e must be raised to produce a given number x and is essential in converting between multiplicative and additive scales in processes like continuous growth.

Exponential Relationships

The exponential function and its inverse, the natural logarithm, are key in solving equations of the form ln(x) = y, which translates directly to x = e^y. This relationship underpins many models of growth and decay in science and finance, helping to translate constant rates of change into mathematical expressions.

Unit Conversion and Scale Representation

Converting units is critical when dealing with real-world representations of mathematical functions on graphs. This involves translating distances measured in one unit (such as centimeters on a coordinate grid) into another unit (such as miles) using known conversion factors, ensuring that abstract numerical results are understood in practical, everyday terms.

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