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Chapter 9: Problem 24

Solve. Where appropriate, include approximations to three decimal places. $$ 7.2^{x}-65=0 $$

Short Answer

Expert verified
x \approx 2.115

Step by step solution

01

- Isolate the exponential term

Add 65 to both sides of the equation to isolate the exponential term.\[ 7.2^{x} - 65 + 65 = 0 + 65 \]This simplifies to:\[ 7.2^{x} = 65 \]
02

- Apply the logarithm

Take the natural logarithm of both sides to remove the exponent. This will give us:\[ \text{ln}(7.2^{x}) = \text{ln}(65) \]
03

- Utilize the power rule of logarithms

Apply the power rule of logarithms, which states that \( \text{ln}(a^b) = b \text{ln}(a) \). This simplifies our equation to:\[ x \text{ln}(7.2) = \text{ln}(65) \]
04

- Solve for x

Divide both sides of the equation by \( \text{ln}(7.2) \):\[ x = \frac{\text{ln}(65)}{\text{ln}(7.2)} \]Using a calculator, we find:\[ x \approx \frac{4.174387}{1.974082} \approx 2.115 \]

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Key Concepts

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

logarithms
Logarithms are the inverse operations of exponentiation. In simpler terms, logarithms help us find the exponent that a base number must be raised to in order to get another number. For example, when you see \(\text{log}_2 8\), it asks, 'To what power must 2 be raised to get 8?' The answer is 3, because \(2^3 = 8\). Logarithms are particularly useful for solving equations where the unknown variable is an exponent. They compress exponential growth and make it more manageable for calculations.
natural logarithm
The natural logarithm, denoted as \( \ln\), is a special type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. Natural logarithms simplify many equations in mathematics and science. For example, in our exercise, we used the natural logarithm to help isolate the variable \(x\). The natural logarithm of a number \(y\), denoted as \( \ln(y) \), tells us the power to which \(e\) must be raised to obtain \(y \).
power rule of logarithms
The power rule of logarithms is a property that allows us to manage exponents within logarithmic expressions. It states: \[ \text{ln}(a^b) = b \text{ln}(a) \]. This rule simplifies the handling of equations with exponents. In our exercise, the rule helped convert \( \text{ln}(7.2^x) \) to \( x \text{ln}(7.2) \). This step is crucial because it lets us move the variable out of the exponent, making the equation easier to solve.
approximations
Approximations help us get practical solutions when exact values are complex or impossible to calculate. For instance, logarithms and natural logarithms often result in irrational numbers鈥攏umbers that cannot be expressed precisely as fractions. Using calculators or software, we approximate these values to a certain number of decimal places. In the given problem, after applying the necessary logarithmic operations, we approximated the solution to three decimal places, finding \(x \approx 2.115 \). These approximations are particularly useful in real-life situations where slight variations are acceptable.

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Most popular questions from this chapter

The average price paid for a Super Bowl ticket has increased exponentially from 12 dollars in 1967 to 2670 dollars in 2015 Data: DallasNews.com; seatgeek.com a) Find the exponential growth rate \(k,\) and write an equation for an exponential function that can be used to predict the average price paid for a Super Bowl ticket \(t\) years after 1967 b) Estimate the year in which the average price paid for a Super Bowl ticket will reach 5000 dollars.

The Richter scale, developed in \(1935,\) has been used for years to measure earthquake magnitude. The Richter magnitude \(m\) of an earthquake is given by $$ m=\log \frac{A}{A_{0}} $$ where \(A\) is the maximum amplitude of the earthquake and \(A_{0}\) is a constant. What is the magnitude on the Richter scale of an earthquake with an amplitude that is a million times \(A_{0} ?\)

For each function given below, (a) determine the domain and the range, (b) set an appropriate window, and (c) draw the graph. Graphs may vary, depending on the scale used. $$ f(x)=x \ln (x-2.1) $$

As of May 2016 , the highest price paid for a painting was 300 million dollars, paid in 2015 for Willem de Kooning's "Interchange." The same painting was purchased for 20.6 million dollars in \(1989 .\) Data: wsj. com, \(2 / 25 / 16\) a) Find the exponential growth rate \(k,\) and determine the exponential growth function that can be used to estimate the painting's value \(V(t),\) in millions of dollars, \(t\) years after \(1989 .\) b) Estimate the value of the painting in 2025 c) What is the doubling time for the value of the painting? d) How many years after 1989 will it take for the value of the painting to reach 1 billion dollars?

Solve \(\log _{4}(3 x-2)=2\)
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