Problem 18 Solve the following equations fo... [FREE SOLUTION]

91影视

Calculus and Its Applications

Larry J. Goldstein, David I. Schneider, Nakhle H. Asmar

$Math Studyset 91影视 Explanations$ Math

13 Edition

Chapter 4: Problem 18

Solve the following equations for $x$ $10^{-x}=10^{2}$

Short Answer

Expert verified

x = -2

Step by step solution

Understand the equation

The given equation is an exponential equation: $10^{-x} = 10^2$

Equate the exponents

Since the bases are the same (both 10), the exponents must be equal. So, $ -x = 2 $

Solve for x

To isolate the variable x, multiply both sides of the equation by -1: $x = -2 $

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Key Concepts

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

Equating Exponents

When solving exponential equations like the one given, it鈥檚 crucial to recognize that if the bases are equal, we can directly equate the exponents. In the exercise, we started with:

$ 10^{-x} = 10^2 $

Here, the base of the exponentials on both sides is 10. Since the bases are identical, we set the exponents equal to each other:

$ -x = 2 $

This step simplifies solving the equation since we change the more complex exponential form into a simpler form involving just the exponents.

Solving for Variables

Once we have equated the exponents, our next task is to isolate the variable. In our example, we have:

$ -x = 2 $

To solve for x, we need to get x alone on one side of the equation. We can do this by considering the inverse operation that will isolate x. Since x is currently multiplied by -1 (as indicated by the negative sign), we multiply both sides of the equation by -1. This gives us:

$ x = -2 $

By performing the inverse operation, we effectively solve for x, giving us the solution to the original equation.

Exponential Functions

Exponential functions are a type of function where the variable appears in the exponent. They are written in the form: $ f(x) = a \cdot b^x $, where a is a constant, b is the base, and x is the exponent. In the problem we solved:

$ 10^{-x} = 10^2 $

The base of the exponent, 10, is crucial because it allows us to equate the exponents directly. Understanding how exponential functions work and how to manipulate them is key to solving these equations.

Exponential functions have unique properties such as:

Their rapid growth or decay, depending on the base being greater than or less than 1.
They model many real-world phenomena like population growth, radioactive decay, and interest calculations.

Recognizing and working with exponential functions is essential in both mathematical and practical applications.

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91影视

Solve the following equations for \(x\) \(10^{-x}=10^{2}\)

Short Answer

Step by step solution

Understand the equation

Equate the exponents

Solve for x

Key Concepts

Equating Exponents

Solving for Variables

Exponential Functions

One App. One Place for Learning.

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