Solve the following exponential equation. 2 = 10 Select the correct choice below and, if necessary, fill in the answer box. O A. X = (Round to three decimal places as needed. Use a comma to separate answers as needed.) B. The solution is not a real number.
Solve the following exponential equation. 2 = 10 Select the correct choice below and, if necessary, fill in the answer box. O A. X = (Round to three decimal places as needed. Use a comma to separate answers as needed.) B. The solution is not a real number.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter5: Exponential And Logarithmic Functions
Section5.2: Applications Of Exponential Functions
Problem 35E
Related questions
Question
![### Solving Exponential Equations
#### Problem Statement:
Solve the following exponential equation:
\[ 2^x = 10 \]
#### Solution Choices:
Please select the correct choice below and, if necessary, fill in the answer box.
- **Option A:**
\[
x = \_\_\_\_
\]
*(Round to three decimal places as needed. Use a comma to separate answers as needed.)*
- **Option B:**
\[
\text{The solution is not a real number.}
\]
#### Explanation:
In this problem, you need to solve the exponential equation \( 2^x = 10 \). You have two options to choose from, and if the answer is a numerical value, it must be rounded to three decimal places.
*Note:* To solve the equation \( 2^x = 10 \), you can use logarithms. Taking the logarithm of both sides of the equation with a common base (such as base 10 or base e) might help in finding the value of \( x \).
1. **Taking the natural logarithm (ln):**
\[
\ln(2^x) = \ln(10)
\]
Using the property of logarithms, \(\ln(a^b) = b \cdot \ln(a)\), we get:
\[
x \cdot \ln(2) = \ln(10)
\]
Solving for \( x \):
\[
x = \frac{\ln(10)}{\ln(2)}
\]
2. **Taking the common logarithm (log):**
\[
\log(2^x) = \log(10)
\]
Using the property of logarithms, \(\log(a^b) = b \cdot \log(a)\), we get:
\[
x \cdot \log(2) = \log(10)
\]
Solving for \( x \):
\[
x = \frac{\log(10)}{\log(2)}
\]
Calculating the value using a calculator:
\[
x = \frac{\log(10)}{\log(2)} \approx \frac{1}{0.301} \approx 3.322
\]
Therefore, the correct choice is:
- **Option](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96f2e08c-aa5d-4493-8570-9d5ecf302bf0%2Fce3a6ff8-4e28-4ff4-a340-77d22ed45081%2F2r82ior_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations
#### Problem Statement:
Solve the following exponential equation:
\[ 2^x = 10 \]
#### Solution Choices:
Please select the correct choice below and, if necessary, fill in the answer box.
- **Option A:**
\[
x = \_\_\_\_
\]
*(Round to three decimal places as needed. Use a comma to separate answers as needed.)*
- **Option B:**
\[
\text{The solution is not a real number.}
\]
#### Explanation:
In this problem, you need to solve the exponential equation \( 2^x = 10 \). You have two options to choose from, and if the answer is a numerical value, it must be rounded to three decimal places.
*Note:* To solve the equation \( 2^x = 10 \), you can use logarithms. Taking the logarithm of both sides of the equation with a common base (such as base 10 or base e) might help in finding the value of \( x \).
1. **Taking the natural logarithm (ln):**
\[
\ln(2^x) = \ln(10)
\]
Using the property of logarithms, \(\ln(a^b) = b \cdot \ln(a)\), we get:
\[
x \cdot \ln(2) = \ln(10)
\]
Solving for \( x \):
\[
x = \frac{\ln(10)}{\ln(2)}
\]
2. **Taking the common logarithm (log):**
\[
\log(2^x) = \log(10)
\]
Using the property of logarithms, \(\log(a^b) = b \cdot \log(a)\), we get:
\[
x \cdot \log(2) = \log(10)
\]
Solving for \( x \):
\[
x = \frac{\log(10)}{\log(2)}
\]
Calculating the value using a calculator:
\[
x = \frac{\log(10)}{\log(2)} \approx \frac{1}{0.301} \approx 3.322
\]
Therefore, the correct choice is:
- **Option
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