/*! 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 18 Solve the equation. \(7^{2 x-3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 18

Solve the equation. \(7^{2 x-3}=\left(\frac{1}{49}\right)^{x+1}\)

Short Answer

Expert verified
x = \(\frac{1}{4}\)

Step by step solution

01

- Rewrite the equation with common bases

Express the equation with a common base. Notice that 49 can be written as a power of 7: 49 = 7^2. Thus, \( \frac{1}{49} = \frac{1}{7^2} = 7^{-2} \). Rewriting the original equation, we get \( 7^{2x-3} = (7^{-2})^{x+1} \).
02

- Simplify the right-hand side

Apply the power rule \((a^m)^n = a^{m \cdot n} \) to simplify the right-hand side: \( (7^{-2})^{x+1} = 7^{-2(x+1)} \).
03

- Set the exponents equal to each other

Since the bases are now the same, set the exponents equal to each other: \( 2x - 3 = -2(x + 1) \).
04

- Solve for x

Solve the equation for x: \( 2x - 3 = -2x - 2 \). Combine like terms: \( 2x + 2x = -2 + 3 \), which simplifies to \( 4x = 1 \). Finally, divide by 4: \( x = \frac{1}{4} \).

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.

Common Bases
When solving exponential equations, it's often useful to rewrite both sides of the equation using the same base. This makes it easier to compare and manipulate the exponents. In our exercise, we started with the equation
\[ 7^{2x-3} = \frac{1}{49}^{x+1} \]
We recognized that 49 is a power of 7, specifically 49 = 7^2. This allowed us to rewrite the right side as
\[ \frac{1}{49} = \frac{1}{7^2} = 7^{-2} \]
Now, the equation becomes
\[ 7^{2x-3} = (7^{-2})^{x+1} \]
Using common bases helps simplify the equation and is a crucial step in solving exponential equations.
Power Rule
The power rule for exponents states that \[ (a^m)^n = a^{m \times n} \]
This rule helps us simplify expressions when an exponent is raised to another exponent. In our equation, after rewriting the equation with common bases, we had
\[ 7^{2x-3} = (7^{-2})^{x+1} \]
Applying the power rule, we get
\[ (7^{-2})^{x+1} = 7^{-2 \times (x+1)} = 7^{-2x - 2} \]
This step simplifies the right-hand side, making it easier to set up the equation for the next move.
Simplifying Exponents
Once you have the equation with common bases and applied the power rule, the next step involves simplifying the exponents. This is done by setting the exponents equal to one another and solving for the variable. Given the equation
\[ 7^{2x-3} = 7^{-2x-2} \]
We can now set the exponents equal because the bases are the same:
\[ 2x-3 = -2(x+1) \]
Simplifying the equation helps us isolate the variable and find its value.
Solving for Variables
This final step involves solving the equation derived from setting the exponents equal. Given:
\[ 2x - 3 = -2(x + 1) \]
We first distribute the -2 on the right-hand side:
\[ 2x - 3 = -2x - 2 \]
Next, we combine like terms:
\[ 2x + 2x = -2 + 3 \]
This simplifies to:
\[ 4x = 1 \]
Finally, we solve for x by dividing both sides by 4:
\[ x = \frac{1}{4} \]
By breaking each step down, we can solve the equation systematically and accurately.

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

Solve for the indicated variable. \(A=P e^{r t}\) for \(r\) (used in finance)

Compare the graphs of the functions. $$ Y_{1}=\ln (2 x) \quad \text { and } \quad Y_{2}=\ln 2+\ln x $$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{8}(3 y-5)+10=12\)

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 640 \\ \hline 20 & 530 \\ \hline 40 & 430 \\ \hline 50 & 360 \\ \hline 80 & 210 \\ \hline 100 & 90 \\ \hline \end{array} $$

The number of computers \(N(t)\) (in millions) infected by a computer virus can be approximated by $$N(t)=\frac{2.4}{1+15 e^{-0.72 t}}$$ where \(t\) is the time in months after the virus was first detected. a. Determine the number of computers initially infected when the virus was first detected. b. How many computers were infected after 6 months? Round to the nearest hundred thousand. c. Determine the amount of time required after initial detection for the virus to affect 1 million computers. Round to 1 decimal place. d. What is the limiting value of the number of computers infected according to this model?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.