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Chapter 5: Problem 10

In Exercises 9-20, solve for \(x\). $$ 3^{x}=243 $$

Short Answer

Expert verified
The solution for \(x\) is 5.

Step by step solution

01

Rewrite 243

Start by rewriting the number 243 as the power of 3 because the left side of the equation has 3 as the base. In fact, \(243 = 3^5\). So the equation now looks like: \(3^x = 3^5\).
02

Solve for x

Given that the bases are the same, we can equate the exponents to find the value of \(x\). So, \(x = 5\).

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Key Concepts

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

Base of an Equation
The base of an equation in exponential problems refers to the number that is being multiplied by itself a certain number of times, indicated by the exponent. In the equation given, \(3^x = 243\), the base is \(3\) on the left side of the equation. When solving exponential equations, it is crucial to express all terms with the same base to simplify the process. This is why we rewrote \(243\) as \(3^5\). By simplifying numbers to the same base, the problem becomes easier to handle and the path to the solution is clearer. Understanding and manipulating the base of the equation is the key first step in solving such exercises.
Equating Exponents
Equating exponents is a method used in solving exponential equations that have the same base. Once the equation \(3^x = 3^5\) is established, you can simply set the exponents equal to each other, since the bases are already equal. This results in \(x = 5\). This step leverages the property of exponential functions which states that if \(a^m = a^n\), then \(m = n\) provided \(a\) is a positive number other than one. Equating exponents simplifies the problem into a basic algebraic equation, making it straightforward to solve. This understanding allows us to solve the equation without directly calculating powers, saving time and effort.
Power of a Number
The power of a number is expressed as \(a^b\), where \(a\) is the base and \(b\) is the exponent indicating how many times the base is multiplied by itself. For example, \(3^5 = 243\), because \(3\) multiplied by itself five times equals \(243\). Recognizing the power of a number helps in breaking down seemingly complex equations like the one in this exercise, where rewriting \(243\) as \(3^5\) was crucial. Understanding how to express numbers as powers is integral to solving equations involving exponentials. It transforms a standalone number into an understandable component of exponential notation, allowing for simpler calculations and solutions.

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Most popular questions from this chapter

In Exercises 45-52, write the logarithmic equation in exponential form. \(\ln \frac{1}{2}=-0.693 \ldots\)

\(9^{\log _{9} 15}\)

\(\log (2 x+1)=\log 15\)

\(f(x)=-\log _{6}(x+2)\)

\(f(x)=e^{x}, \quad g(x)=\ln x\)
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