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
![**Solving Exponential Equations**
To solve the given exponential equation, follow the steps below:
Equation:
\[ 3^{5x + 2} = 24 \]
Step 1: Take the natural logarithm (ln) on both sides of the equation to make use of the properties of logarithms:
\[ \ln(3^{5x + 2}) = \ln(24) \]
Step 2: Use the power rule of logarithms, which states \(\ln(a^b) = b \ln(a)\):
\[ (5x + 2) \ln(3) = \ln(24) \]
Step 3: Isolate the term containing the variable \(x\). First, divide both sides of the equation by \(\ln(3)\):
\[ 5x + 2 = \frac{\ln(24)}{\ln(3)} \]
Step 4: Solve for \(x\):
\[ 5x = \frac{\ln(24)}{\ln(3)} - 2 \]
\[ x = \frac{\frac{\ln(24)}{\ln(3)} - 2}{5} \]
The approximate solution for \(x\) can be found using a calculator to compute the natural logarithms:
\[ x \approx \frac{\frac{3.178}{1.098} - 2}{5} \approx \frac{2.895 - 2}{5} \approx \frac{0.895}{5} \approx 0.179 \]
Thus, \(x \approx 0.179\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50351a65-5e52-4545-9e5a-caf58e03b0d0%2Fb792a3df-12f6-41bf-90de-03be04c908cf%2Fcmric0f_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Exponential Equations**
To solve the given exponential equation, follow the steps below:
Equation:
\[ 3^{5x + 2} = 24 \]
Step 1: Take the natural logarithm (ln) on both sides of the equation to make use of the properties of logarithms:
\[ \ln(3^{5x + 2}) = \ln(24) \]
Step 2: Use the power rule of logarithms, which states \(\ln(a^b) = b \ln(a)\):
\[ (5x + 2) \ln(3) = \ln(24) \]
Step 3: Isolate the term containing the variable \(x\). First, divide both sides of the equation by \(\ln(3)\):
\[ 5x + 2 = \frac{\ln(24)}{\ln(3)} \]
Step 4: Solve for \(x\):
\[ 5x = \frac{\ln(24)}{\ln(3)} - 2 \]
\[ x = \frac{\frac{\ln(24)}{\ln(3)} - 2}{5} \]
The approximate solution for \(x\) can be found using a calculator to compute the natural logarithms:
\[ x \approx \frac{\frac{3.178}{1.098} - 2}{5} \approx \frac{2.895 - 2}{5} \approx \frac{0.895}{5} \approx 0.179 \]
Thus, \(x \approx 0.179\).
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