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Chapter 1: Problem 9

In Exercises 9 through \(16,\) solve for \(x\). $$ 3^{x}=9 $$

Short Answer

Expert verified
The solution is \(x = 2\).

Step by step solution

01

Express 9 as a Power of 3

First, recognize that 9 can be expressed as a power of 3. Since \(9 = 3^2\), you can rewrite the equation using the same base, which gives us \(3^x = 3^2\).
02

Set the Exponents Equal to Each Other

Since the bases are the same, we can set the exponents equal to each other to solve for \(x\). This simplifies the equation to \(x = 2\).
03

Write the Solution

Having simplified the equation to \(x = 2\), this is the value of \(x\) that solves the original equation. You can verify this by substituting back into the original equation to see that both sides agree.

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Key Concepts

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

Understanding Exponentiation
Exponentiation is a fundamental concept in mathematics, where a number, called the base, is raised to a power, represented by an exponent. It involves repeated multiplication of the base.
  • For example, in the expression \(3^2\), the number 3 is the base, and the exponent is 2. This means the base is multiplied by itself: \(3 \times 3 = 9\).
  • Exponentiation is denoted as \(a^b\), where \(a\) is the base and \(b\) is the exponent.
Exponentiation simplifies complex multiplication tasks. Instead of multiplying a number several times, exponentiation allows us to express it neatly using powers, which is particularly effective when dealing with larger numbers or more complex equations.
The Role of Algebra in Solving Exponential Equations
Algebra provides the tools and rules for solving many types of equations, including exponential ones. When faced with an equation like \(3^x = 9\), algebra helps us manipulate expressions to find the unknown variable.
  • The first step, expressing both sides of the equation with the same base, is a key algebraic technique. Here, recognizing that 9 can be rewritten as \(3^2\) simplifies the problem.
  • Once the equation has the form \(3^x = 3^2\), algebra allows us to set the exponents equal when the bases match, leading directly to the solution \(x = 2\).
Algebraic manipulation thus becomes instrumental in unraveling the equation, with each step founded on logical rules and properties of mathematics.
Teaching and Learning Mathematics
Mathematics education focuses not only on learning procedures but also on understanding the underlying concepts. This approach helps students apply their knowledge broadly and flexibly.
  • Conceptual understanding is vital for solving problems effectively. Students should learn why the steps in solving an exponential equation work, not just how to execute them.
  • Practice with feedback is essential for mastery. By attempting various problems and receiving guidance, learners develop stronger skills and confidence in handling similar equations.
Understanding the process rather than memorizing steps prepares students for advanced mathematical challenges, fostering deeper appreciation and robust problem-solving abilities.

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Most popular questions from this chapter

Usher and coworkers \({ }^{18}\) developed a mathematical model for certain farm woodlands, described by the equation \(S(A)=1.81 A^{0.284},\) where \(S\) is the number of species of plants and \(A\) is the area of woodland in square meters. Graph. Explain what \(S(118)\) means. What happens to the number of species if the area is reduced by one half?

Find the present value of the given amounts \(F\) with the indicated annual rate of return \(r,\) the number of years \(t,\) and the indicated compounding. \(F=\$ 10,000, r=10 \%, t=20,\) compounded (a) annually, (b) quarterly, (c) daily, (d) continuously.

If an object is initially at a height above the ground of \(s_{0}\) feet and is thrown straight upward with an initial velocity of \(v_{0}\) feet per second, then from physics it can be shown the height in feet above the ground is given by \(s(t)=-16 t^{2}+v_{0} t+s_{0},\) where \(t\) is in seconds. Find how long it takes for the object to reach maximum height. Find when the object hits the ground.

\(\begin{array}{lllll}\text { Growth } & \text { Rates of } & \text { Cutthroat } & \text { Trout Ruzycki } & \text { and }\end{array}\) coworkers \(^{75}\) studied the growth habits of cutthroat trout in Bear Lake, Utah-Idaho. The mathematical model they created was given by the equation \(L(t)=650(1-\) \(e^{-0.25[t+0.50]}\), where \(t\) is age in years and \(L\) is length in millimeters. Graph this equation. What seems to be happening to the length as the trout become older? Ruzycki and coworkers also created a mathematical model that connected length with weight and was given by the equation \(W(L)=1.17 \times 10^{-5} \cdot L^{2.93},\) where \(L\) is length in millimeters and \(W\) is weight in grams. Find the length of a 10-year old cutthroat trout. Find the weight of a 10 -year old cutthroat trout. Find \(W\) as a function of \(t\).

Equations In 1996 Rogers and Akridge \(^{38}\) of Purdue University studied fertilizer plants in Indiana. For a typical small-sized plant they estimated fixed costs at \(\$ 235,487\) and estimated that it cost \(\$ 206.68\) to produce each ton of fertilizer. The plant sells its fertilizer output at \(\$ 266.67\) per ton. Find the cost, revenue, and profit equations.
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