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 $f(x)=\log x$ 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 the result to the nearest mile. Why is this result so much larger than the result in Exercise $35 ?$

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 $f(x)=\log x$ 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 the result to the nearest mile. Why is this result so much larger than the result in Exercise $35 ?$

Key Concepts

Magnitude Differences in Continuous Versus Discrete Settings

The result being much larger than in another related exercise (such as Exercise 35) can be attributed to the nature of continuous growth versus more discrete or less uniformly scaled growth models. In continuous processes, the exponential relationship means that even modest increases in the logarithmic output correspond to extremely large multiplicative increases in the original quantity, leading to far larger numerical values than might be expected in a discretely scaled or alternative growth context.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It takes an input x and gives the exponent to which a given base must be raised to produce that x. In a coordinate system where both axes have the same unit scale, the logarithmic function grows very slowly compared to a linear function, which means that achieving a high function value, like 12, requires an exponentially large x-value.

Exponential Growth and Its Inversion

Exponential growth is characterized by quantities that increase at rates proportional to their current size. When growth is continuous, the natural logarithm is used to reverse the process and solve for time or rate. In the context of the logarithmic function f(x) = log x, finding the x-value corresponding to a given y-value involves exponentiating that y-value. This is why an output of 12, for example, corresponds to x being an exponential magnitude larger than 12.

Graph Scaling and Unit Conversions

Graph scaling refers to the relationship between units on the graph and the actual numerical values they represent. When plotting functions like f(x) = log x with one centimeter representing one unit along both axes, the visual length required to depict large values of x can be enormous. To convert the required length (determined by the exponential relationship) into familiar units such as miles, one must perform proper unit conversion, highlighting the dramatic differences between linear and logarithmic scales.

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