/*! 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 6 Solve the exponential equation. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 6

Solve the exponential equation. Round to three decimal places, when needed. $$7^{2 x}=49$$

Short Answer

Expert verified
The solution to the equation \(7^{2x} = 49\) is \(x = 1\).

Step by step solution

01

Rewrite the exponential equation

First, the given equation, \(7^{2x} = 49\), can be rewritten as \(7^{2x} = 7^2\) as 49 is the square of 7. By doing this, we can use the properties of the exponential functions to equate the exponents.
02

Equate the exponents

Once the bases are the same, the exponents can be set equal to each other. Thus the equation becomes: \(2x = 2\).
03

Solve for x

Finally, to solve for x, simply divide both sides of the equation by 2: \(x = 2 / 2 = 1\).

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.

Exponential Functions
Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. These functions are written in the form \(a^x\), where \(a\) is a positive constant and \(x\) is the exponent. These functions are characterized by a rapid rate of growth or decay depending on the base value and the sign of the exponent.
With exponential functions, the base determines the nature of the graph:
  • If \(a > 1\), the function is growing, and the graph rises steeply.
  • If \(0 < a < 1\), the function is decaying, and the graph falls sharply.
Understanding exponential functions is crucial as they are commonly used to model real-world scenarios such as population growth, radioactive decay, and interest calculations.
Solving Equations
Solving equations, particularly exponential ones, often involves finding the value of the variable that makes the equation true. The goal is to isolate the variable, often requiring manipulation of the equation using algebra.
In equations involving exponentials, it's important to first ensure the bases are the same if they are not already. For example, in the equation \(7^{2x} = 49\), recognizing that 49 can be written as \(7^2\) allows you to equate the exponents directly.
Here are some steps generally used for solving exponential equations:
  • Rewrite the equation so both sides have the same base.
  • Once the bases match, set the exponents equal to each other.
  • Solve the resulting simple equation for the variable.
By following these steps, solving exponential equations becomes a straightforward process.
Properties of Exponents
Understanding the properties of exponents is key to simplifying and solving exponential equations. These properties provide shortcuts and rules to make calculations easier.
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{mn}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Zero Exponent: \(a^0 = 1\) (where \(a eq 0\))
These properties are useful when working with exponential functions, as they allow you to effectively manipulate expressions to solve equations. For example, knowing \(49 = 7^2\) and rewriting the base this way fundamentally relies on understanding these exponent properties.

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 expression to four decimal places using a calculator. $$\ln \sqrt{2}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln 2 x-\ln \left(x^{2}+1\right)=\ln 1$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 6000 ; r=6.25 \%\)

The value of a 2006 S-type Jaguar is given by the function $$v(t)=43,173(0.8)^{t}$$ where \(t\) is the number of years since its purchase and \(v(t)\) is its value in dollars. (Source: Kelley Blue Book) (a) What was the Jaguar's initial purchase price? (b) What percentage of its value does the Jaguar S-type lose each year? (c) How many years will it take for the Jaguar S-type to reach a value of \(\$ 22,227 ?\)

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 2.75$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.