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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Rewrite equation without logs. Do not solve for x
![**Solving Logarithmic and Quadratic Equations**
In this activity, we will focus on solving logarithmic equations and converting them to quadratic form.
**Given Problem:**
Solve the logarithmic equation. Be sure to justify all steps:
\[ \log_8 x + \log_8 (7x - 1) = 1 \]
**Step-by-Step Solution:**
1. **Combine the logarithms**:
Using the logarithm property: \(\log_a b + \log_a c = \log_a (bc)\):
\[ \log_8 [x(7x - 1)] = 1 \]
2. **Rewrite the equation without logarithms**:
Recall that \(\log_a b = c\) is equivalent to \(a^c = b\):
\[ 8^1 = x(7x - 1) \]
Simplifying the right side:
\[ 8 = x(7x - 1) \]
Which simplifies to:
\[ 8 = 7x^2 - x \]
3. **Rearrange into standard quadratic form**:
\[ 7x^2 - x - 8 = 0 \]
4. **Solve the quadratic equation**:
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 7\), \(b = -1\), and \(c = -8\):
Plug in the values:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-8)}}{2(7)} \]
Simplify under the square root:
\[ x = \frac{1 \pm \sqrt{1 + 224}}{14} \]
\[ x = \frac{1 \pm \sqrt{225}}{14} \]
\[ x = \frac{1 \pm 15}{14} \]
This results in two possible solutions:
\[ x = \frac{16}{14} \]
or
\[ x = \frac{-14}{14} \]
Simplify the fractions:
\[ x = \frac{8}{7} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2107c9d4-03cf-46c4-aa9a-f455311ac0bc%2F2e30f4be-377b-4243-8ec1-55706b2aee71%2Fxb1cldr_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Logarithmic and Quadratic Equations**
In this activity, we will focus on solving logarithmic equations and converting them to quadratic form.
**Given Problem:**
Solve the logarithmic equation. Be sure to justify all steps:
\[ \log_8 x + \log_8 (7x - 1) = 1 \]
**Step-by-Step Solution:**
1. **Combine the logarithms**:
Using the logarithm property: \(\log_a b + \log_a c = \log_a (bc)\):
\[ \log_8 [x(7x - 1)] = 1 \]
2. **Rewrite the equation without logarithms**:
Recall that \(\log_a b = c\) is equivalent to \(a^c = b\):
\[ 8^1 = x(7x - 1) \]
Simplifying the right side:
\[ 8 = x(7x - 1) \]
Which simplifies to:
\[ 8 = 7x^2 - x \]
3. **Rearrange into standard quadratic form**:
\[ 7x^2 - x - 8 = 0 \]
4. **Solve the quadratic equation**:
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 7\), \(b = -1\), and \(c = -8\):
Plug in the values:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-8)}}{2(7)} \]
Simplify under the square root:
\[ x = \frac{1 \pm \sqrt{1 + 224}}{14} \]
\[ x = \frac{1 \pm \sqrt{225}}{14} \]
\[ x = \frac{1 \pm 15}{14} \]
This results in two possible solutions:
\[ x = \frac{16}{14} \]
or
\[ x = \frac{-14}{14} \]
Simplify the fractions:
\[ x = \frac{8}{7} \]
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