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
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### 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
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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