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Chapter 6: Problem 46

Solve each exponential equation. Express irrational solutions in exact form. $$ 5^{-x}=25 $$

Short Answer

Expert verified
x = -2.

Step by step solution

01

- Simplify the Right Side

Rewrite 25 as a power of 5. Since 25 is the same as 5 squared, we can write it as: 25 = 5^2.
02

- Set the Exponents Equal

Since the bases are the same, set the exponents equal to each other: -x = 2.
03

- Solve for x

Solve the equation for x by dividing both sides by -1: x = -2.

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Key Concepts

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

Exponential Equations
Exponential equations are equations where the variable appears in an exponent. These types of equations are crucial in various fields, such as science and engineering.
To solve exponential equations, the key step is to rewrite each side of the equation with the same base if possible.
For example, in the equation given: \[ 5^{-x} = 25 \], we know we can rewrite 25 as a power of 5 because 25 is the same as \(5^2\).
This simplifies the equation significantly.
With the same base on both sides of the equation, we can then set the exponents equal to each other to solve for the variable.
Simplifying Expressions
Simplifying expressions is another important concept when solving exponential equations.
In our exercise, we simplified 25 by recognizing it as \(5^2\).

This step is crucial because it allows us to make both sides of the equation comparable. Let's break this down:
  • Look at the number and check if it can be rewritten as a power of the base (here, the base is 5).
  • Since 25 is a power of 5, we write it as \(25 = 5^2\).

Using these properties to simplify can make solving the equation much easier.
Remember to always express numbers with their respective powers whenever possible.
Solving for Variables
Finally, we solve for the variable once we have made the bases the same.
This involves setting the exponents equal to each other. For example: in the given equation, after simplifying it to \(5^{-x} = 5^2\), we then set: \(-x = 2\).
To find the value of x:
  • Isolate the variable.
  • Solve the equation. In this case, by dividing both sides by -1, we get: \( x = -2\).

Always double-check your work. Verify that your solution satisfies the original equation.
This process ensures that the steps taken were correct and that the solution is accurate.

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Most popular questions from this chapter

Suppose that \(F(x)=\log _{2}(x+1)-3\) (a) What is the domain of \(F ?\) (b) What is \(F(7) ?\) What point is on the graph of \(F ?\) (c) If \(F(x)=-1,\) what is \(x ?\) What point is on the graph of \(F ?\) (d) What is the zero of \(F ?\)

If \(f(x)=5 x^{2}+4 x-8\) and \(g(x)=3 x-1,\) find \((f \circ g)(x)\)

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