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Chapter 4: Problem 80

Solve each equation. Round answers to four decimal places. $$(1.00012)^{365 t}=2.4$$

Short Answer

Expert verified
t \rightarrow 20.1808

Step by step solution

01

- Take the Natural Logarithm of Both Sides

Start by taking the natural logarithm on both sides of the equation to simplify the exponent. This gives: $$\text{ln}((1.00012)^{365 t}) = \text{ln}(2.4)$$
02

- Apply the Power Rule of Logarithms

Use the power rule of logarithms, which states \text{ln}(a^b) = b \text{ln}(a), to move the exponent in front of the logarithm: $$365 t \text{ln}(1.00012) = \text{ln}(2.4)$$
03

- Solve for t

Isolate the variable t by dividing both sides of the equation by 365 \text{ln}(1.00012): $$t = \frac{\text{ln}(2.4)}{365 \text{ln}(1.00012)}$$ Calculate the values of the natural logarithms and then perform the division: $$\text{ln}(2.4) \rightarrow 0.8754687374$$ $$\text{ln}(1.00012) \rightarrow 0.0001199655$$ $$t = \frac{0.8754687374}{365 \times 0.0001199655} \rightarrow t \rightarrow 20.1808$$ Therefore, t \rightarrow 20.1808.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

natural logarithm
A natural logarithm, often abbreviated as \text{ln}, is the logarithm to the base of the mathematical constant e (approximately 2.71828). The natural logarithm is particularly useful in solving exponential equations, as it allows us to linearize the exponent.
When we have an equation like \text{(1.00012)^{365 t} = 2.4}, taking the natural logarithm of both sides transforms the equation into a more manageable form:

\( \text{ln}((1.00012)^{365 t}) = \text{ln}(2.4) \)
This step is crucial because logarithms, particularly natural logarithms, help in simplifying equations where the variable is an exponent.
power rule of logarithms
The power rule of logarithms states that the logarithm of a number raised to a power can be simplified by multiplying the logarithm by that power.
Mathematically, the power rule is written as 鉄�

    \text{ln}(a^b) = b * \text{ln}(a)
    .

For the equation at hand, once we apply the natural logarithm, we use this rule to move the exponent in front:
鉄�
\( \text{ln}((1.00012)^{365 t}) = 365 t \text{ln}(1.00012) \)

This significantly simplifies the equation and allows us to isolate the variable.
isolating the variable
After applying the power rule, the equation becomes:
鉄�
\( 365 t \text{ln}(1.00012) = \text{ln}(2.4) \)
.
The next step is to isolate the variable t. This involves dividing both sides of the equation by 365 * ln(1.00012):
鉄�
\( t = \frac{\text{ln}(2.4)}{365 \text{ln}(1.00012)} \)
.
We then calculate the values for \text{ln}(2.4) and \text{ln}(1.00012).
鉄�
After performing these calculations, the equation simplifies to:
鉄�
\( t \rightarrow 20.1808 \)
.
Isolating the variable in this manner helps us find the specific value of t that satisfies the original equation.
exponential functions
An exponential function is a mathematical function of the form \( f(x) = a * b^x \), where b is a positive real number not equal to 1, and x is the variable. These functions are characterized by their rapid growth (or decay) and are commonly used in various fields like finance, physics, and biology.
In the given problem, our original equation 鉄�
\((1.00012)^{365 t} = 2.4\)
is an example of an exponential function, where the exponent depends on the variable t.
Solving such equations often involves logarithms because they provide a tool to linearize the exponent.
Thus, exponential functions play a critical role in this exercise as they are the basis for using logarithms to find the value of the unknown variable.

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Most popular questions from this chapter

Solve each problem. Because of the Black Death, or plague, the only substantial period in recorded history when the earth's population was not increasing was from 1348 to \(1400 .\) During that period the world population decreased by about 100 million people. Use the exponential model \(P=P_{0} e^{r t}\) and the data from the accompanying table to find the annual growth rate for the period 1400 to 2000 . If the 100 million people had not been lost, then how many people would they have grown to in 600 years using the growth rate that you just found? $$\begin{array}{|c|c|}\hline \text { Year } & \begin{array}{c}\text { World } \\\\\text { Population }\end{array} \\\\\hline 1348 & 0.47 \times 10^{9} \\\1400 & 0.37 \times 10^{9} \\\1900 & 1.60 \times 10^{9} \\\2000 & 6.07 \times 10^{9} \\\\\hline\end{array}$$

Simplify \(\log _{6}(4)+\log _{6}(9)\).

Work in a small group to write a summary (including drawings) of the types of graphs that can be obtained for logarithmic functions of the form \(y=\log _{a}(x)\) for \(a>0\) and \(a \neq 1\).

Solve each problem. Ben Franklin's gift of \(\$ 4000\) to the city of Philadelphia in 1790 was not managed as well as his gift to Boston. The Philadelphia fund grew to only \(\$ 2\) million in 200 years. Find the annual rate compounded continuously that would yield this total value.

Computers per Capita The number of personal computers per 1000 people in the United States from 1990 through 2010 is given in the accompanying table (Consumer Industry Almanac, www.c-i-a.com). a. Use exponential regression on a graphing calculator to find the best- fitting curve of the form \(y=a \cdot b^{x},\) where \(x=0\) corresponds to 1990. b. Write your equation in the form \(y=a e^{e x}.\) c. Assuming that the number of computers per 1000 people is growing continuously, what is the annual percentage rate? d. In what year will the number of computers per 1000 people reach \(1500 ?\) e. Judging from the graph of the data and the curve, does the exponential model look like a good model? $$\begin{array}{|l|c|} \hline \text { Year } & \begin{array}{c} \text { Computers } \\ \text { per 1000 } \end{array} \\ \hline 1990 & 192 \\ 1995 & 321 \\ 2000 & 628 \\ 2005 & 778 \\ 2010 & 932 \\ \hline \end{array}$$
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