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Chapter 3: Problem 1

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$2^{x}=64$$

Short Answer

Expert verified
The solution for the exponential equation \(2^{x}=64\) is \(x=6\).

Step by step solution

01

Rewrite 64 as a power of 2

Since 64 is a power of 2, rewrite it as \(2^{6}\) because \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\). So, now the equation becomes \(2^{x}=2^{6}\).
02

Set the exponents equal

By the rule of exponents, if the bases are the same and the equation is true, then the exponents are equal. So we get \(x = 6\).

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Key Concepts

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

Equating Exponents
Equating exponents is a technique used to solve equations where each side can be expressed as a power of the same base. The fundamental idea is that if the bases are identical, then the exponents must also be the same for the equality to hold true. For instance, if we have the equation \(a^{m} = a^{n}\), and \(a\) is not zero, we can deduce that \(m = n\). This method of equating exponents simplifies complex expressions into more manageable forms.
  • Ensure that both sides of the equation share the same base.
  • Rewrite numbers as powers of the base if necessary, like with the number 64 being written as \(2^6\).
  • Once both sides of the equation have the same base, set the exponents equal to each other to solve for the unknown variable.
This process is especially helpful in problems where direct computation isn't feasible and helps illustrate the relationship between bases and exponents.
Powers of Numbers
A power is the result of multiplying a number by itself a certain number of times. The number that is being multiplied is known as the base, while the number of times the base is used as a factor is called the exponent. For instance, in \(2^6\), the base is 2, and the exponent is 6, indicating that 2 is multiplied by itself six times, resulting in 64.
  • Base: The number being multiplied.
  • Exponent: The number of times the base multiplies itself.
  • Power: The resultant value after the base is raised to the given exponent.
Understanding how to express numbers as powers of another is crucial, especially when solving exponential equations, like recognizing 64 as a power of 2. This skill aids in simplifying problems and making them more manageable.
Exponentiation Rules
Exponentiation rules provide a set of guidelines for working with powers. Understanding these rules is essential for simplifying expressions and solving equations efficiently. Here are some key rules:
  • Product of Powers Rule: When multiplying like bases, add the exponents, as in \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers Rule: When dividing like bases, subtract the exponents, given by \(a^m / a^n = a^{m-n}\).
  • Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
These rules significantly reduce the complexity of working with exponents, making it easier to manipulate and solve equations. By mastering these rules, you'll become more proficient in handling exponential expressions.

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Most popular questions from this chapter

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$ f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12 $$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.

The formula \(A=25.1 e^{0.0187 t}\) models the population of Texas, \(A\), in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?

By 2019 , nearly 1 dollar out of every 5 dollars spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product (GDP) going toward health care from 2007 through 2014 , with a projection for 2019.(GRAPH CAN'T COPY). The data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2009 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.5 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f. $$f(x)=\log x, g(x)=\log (x-2)+1$$
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