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Chapter 1: Problem 17

Solve the given problem for \(x,\) if possible. If a problem cannot be solved, explain why. $$ 5^{x}=1 $$

Short Answer

Expert verified
The solution is \(x = 0\).

Step by step solution

01

Understand the Equation Structure

The equation given is \(5^{x} = 1\). This equation represents an exponential equation where the base is 5 and it is raised to the power of \(x\). Our goal is to find the value of \(x\) that satisfies this equation.
02

Recognize Properties of Exponents

Recall that any non-zero number raised to the power of zero equals 1, as in the property \(a^0 = 1\), given \(aeq0\). This means that for any non-zero base, if the result is 1, the exponent has to be 0.
03

Apply the Exponent Property to Find x

Using the property from Step 2, we can conclude that since the equation is \(5^{x} = 1\), the value of \(x\) must be 0. So, \(x = 0\) is the solution to the equation, as \(5^0 = 1\).

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Key Concepts

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

Properties of Exponents
Exponents are an essential part of algebra, providing a shorthand method to represent repeated multiplication of a number by itself. There are several key properties of exponents that help simplify and solve equations:
  • Product of Powers: When multiplying two expressions with the same base, you can add the exponents: \(a^m \cdot a^n = a^{m+n}\).
  • Quotient of Powers: When dividing two expressions with the same base, subtract the exponent of the denominator from that of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power of a Power: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
  • Power of a Product: To raise a product to a power, apply the exponent separately to each factor: \((ab)^m = a^m \cdot b^m\).
  • Zero Exponent Rule: Any non-zero base raised to the power of zero equals 1: \(a^0 = 1\) if \(a eq 0\).
Understanding these properties allows you to manipulate and simplify expressions, especially when solving equations involving exponents.
Solving Exponential Equations
Exponential equations can seem tricky at first, but they become straightforward once you understand a few key strategies. An exponential equation features a variable in the exponent, making techniques to solve them unique.
  • Equal Bases Strategy: One common method is to rewrite each side of the equation with the same base if possible. Once they are the same, set the exponents equal to each other. For instance, if you have \(3^{x} = 27\), you can rewrite 27 as \(3^3\), then set \(x = 3\).
  • Logarithms: When it's not possible to equate the bases, logarithms are useful. Taking the log of both sides helps bring the variable down from the exponent, allowing you to solve for it using algebraic methods.
In the exercise provided, we used an understanding of properties, particularly the zero exponent rule, which made solving \(5^x = 1\) straightforward.
Zero Exponent Rule
The Zero Exponent Rule is a fundamental property of exponents that states any non-zero number raised to the power of zero is 1. This rule is derived from the pattern observed when reducing powers.Let's break this down:
  • Consider decreasing powers of a number: \(a^3 = a \cdot a \cdot a\), \(a^2 = a \cdot a\), \(a^1 = a\).
  • Notice each step involves dividing by the base \(a\), thus \(\frac{a^1}{a} = a^0 = 1\).
This understanding can be directly applied to solving the original problem of \(5^x = 1\). According to this rule, the only solution is when the exponent is 0, leading us to the conclusion that \(x = 0\). This simple yet powerful rule is invaluable in solving exponential equations effectively.

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Most popular questions from this chapter

Let \(f(x)=x^{3}, g(x)=x+4,\) and \(h(x)=\sin (x) .\) Each of exercise \(5-8\) is some combination of \(f(x), g(x),\) and \(h(x) .\) Determine the type of combination and write it using function notation. For example, \(x^{3}+x+4\) is the addition of \(f(x)\) and \(g(x)\) and can be written as \(f(x)+g(x)\). $$ \frac{x^{3}}{\sin (x)} $$

Evaluate the described mathematical statement, or determine how the described changes affect other variables in the statement as appropriate. The Reynold's number, which helps identify whether or not a fluid flow is turbulent, is given by \(\mathrm{Re}=\frac{\rho \mathrm{uD}}{\mu} .\) If \(\rho, \mathrm{u},\) and \(\mathrm{D}\) are held constant while \(\mu\) increases, does Re increase, decrease, or stay the same?

In exercises \(11-17,\) simplify the given term and write your answer without negative exponents. $$ \left(\frac{-3 s^{2 / 3} t^{2}}{4 s^{3} t^{5 / 3}}\right)^{3} $$

In exercises \(13-20\), factor each function completely. $$ g(x)=4 x^{2}+4 x+1 $$

In exercises \(11-17,\) simplify the given term and write your answer without negative exponents. $$ \left(\frac{-2 x^{2 / 3} y^{2}}{x^{-2} y^{1 / 2}}\right)^{6} $$
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