Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![**Mathematics Challenge: Solving Exponential Equations**
Equation to Solve:
\[ 4^{x-1} + 3(2^{x+1}) = 13 \]
**Explanation:**
This is an example of an exponential equation where you have to solve for the variable \( x \). The equation includes base powers of 4 and 2. Here are a few steps and tips that might help:
1. **Rewrite the Exponents:** Start by expressing terms like \( 4^{x-1} \) and \( 2^{x+1} \) in a more manageable form if possible. Remember that \( 4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)} \).
2. **Isolate Terms:** Try to isolate terms involving the variable. For example, simplify or rewrite terms to factor or combine them.
3. **Solve for \( x \):** Look for ways to simplify the equation so you can solve for \( x \), either through algebraic manipulation or using logarithms.
4. **Check Your Solution:** Once you find a value for \( x \), substitute it back into the original equation to ensure it satisfies the equation.
This problem tests your understanding of exponential functions and how to manipulate them algebraically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7bfde65-f228-4aa5-b485-e443d3b4afbd%2Fe33c4feb-260a-4495-91c9-42a5f05a94cc%2Fxvvk6qw.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Challenge: Solving Exponential Equations**
Equation to Solve:
\[ 4^{x-1} + 3(2^{x+1}) = 13 \]
**Explanation:**
This is an example of an exponential equation where you have to solve for the variable \( x \). The equation includes base powers of 4 and 2. Here are a few steps and tips that might help:
1. **Rewrite the Exponents:** Start by expressing terms like \( 4^{x-1} \) and \( 2^{x+1} \) in a more manageable form if possible. Remember that \( 4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)} \).
2. **Isolate Terms:** Try to isolate terms involving the variable. For example, simplify or rewrite terms to factor or combine them.
3. **Solve for \( x \):** Look for ways to simplify the equation so you can solve for \( x \), either through algebraic manipulation or using logarithms.
4. **Check Your Solution:** Once you find a value for \( x \), substitute it back into the original equation to ensure it satisfies the equation.
This problem tests your understanding of exponential functions and how to manipulate them algebraically.
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