/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Solve the following equations fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

Solve the following equations for \(x\) \(5^{2 x}=5^{2}\)

Short Answer

Expert verified
x = 1

Step by step solution

01

Understand the Equation

The given equation is \(5^{2 x} = 5^{2}\). Since the bases are the same, we can set the exponents equal to each other.
02

Set Exponents Equal

Set the exponents equal to each other: \(2x = 2\).
03

Solve for x

Solve for \(x\) by dividing both sides by 2: \(\frac{2x}{2} = \frac{2}{2}\), which simplifies to \(x = 1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Exponents
Exponents are a way to represent repeated multiplication of the same number. For example, in the expression \(5^3\), the number 5 is the base, and 3 is the exponent. This tells us to multiply 5 by itself three times: \[5^3 = 5 \times 5 \times 5 = 125\].
By understanding exponents, we can simplify complex multiplication and better grasp the relationships between numbers.
  • In the problem \5^{2x} = 5^2\, having the same base (5) makes it easier to solve for x.
  • When the bases are equal, you can set the exponents equal to each other to simplify the equation. In this case, it becomes \2x = 2\.
Solving Exponential Equations
Exponential equations are equations where unknown variables appear as exponents. For example, in the equation \5^{2x} = 5^2\, 2x is the exponent of the base 5.
  • To solve exponential equations, it is helpful if the bases on both sides of the equation are the same.
  • When the bases are the same, you can simply set the exponents equal to each other and solve for the unknown variable.

In the provided exercise, since both exponents share a common base (5), we can set \2x = 2\. Then, to find the value of x, we divide both sides by 2: \[ \frac{2x}{2} = \frac{2}{2} \] This simplifies to x = 1.
Fundamental Algebra Concepts
Algebra involves using symbols and letters to represent numbers and quantities in expressions and equations.
In the given problem, our task is to solve the equation \(5^{2x} = 5^2 \) for x. Here's how algebraic concepts help:
  • First, we recognize that since the bases are the same (both are 5), we can set the exponents equal. This is a fundamental property of exponents.
  • Next, the resulting equation \(2x = 2\) is a simple linear equation. By using basic algebraic methods like division, we solve for x.

It means we apply properties of exponents and linear equations to find the value of x. The basic steps are straightforward: here, dividing both sides of \(2x = 2\) by 2 gives us \x = 1\.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the given expression. Use \(\ln 2=.69\) and \(\ln 3=1.1\). (a) \(\ln \frac{1}{6}\) (b) \(\ln \frac{2}{9}\) (c) \(\ln \frac{1}{\sqrt{2}}\)

Evaluate the given expression. Use \(\ln 2=.69\) and \(\ln 3=1.1\). (a) \(\ln 100-2 \ln 5\) (b) \(\ln 10+\ln \frac{1}{5}\) (c) \(\ln \sqrt{108}\)

Differentiate the following functions. \(y=\frac{\ln x}{2}\)

Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function \(f(x)=26.48-14.09 \ln x\) gives the number of mosquito bites received when the concentration was \(x\) percent. \([\) Note: The answers to parts \((\mathrm{b})-(\mathrm{e})\) can be obtained either algebraically or from the graphs. You might consider trying both methods.] (Source: Journal of Medical Entomology.) (a) Graph \(f(x)\) and \(f^{\prime}(x)\) for \(0

Differentiate the following functions. \(y=(\ln x)^{2}+\ln \left(x^{2}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.