Answer :
Sure, let's start by solving the given equation step-by-step:
[tex]\[ 8e^2 - 5 = 0 \][/tex]
### Step 1: Solving for [tex]\( e^2 \)[/tex]
[tex]\[ 8e^2 = 5 \][/tex]
[tex]\[ e^2 = \frac{5}{8} \][/tex]
### Step 2: Taking the natural logarithm on both sides
[tex]\[ \ln(e^2) = \ln\left(\frac{5}{8}\right) \][/tex]
Since [tex]\( \ln(e^2) = 2\ln(e) \)[/tex], we get:
[tex]\[ 2 \ln(e) = \ln\left(\frac{5}{8}\right) \][/tex]
[tex]\[ \ln(e) = \frac{1}{2}\ln\left(\frac{5}{8}\right) \][/tex]
Now let's rewrite [tex]\( \ln\left(\frac{5}{8}\right) \)[/tex] using logarithm properties:
[tex]\[ \ln\left(\frac{5}{8}\right) = \ln(5) - \ln(8) \][/tex]
So:
[tex]\[ \ln(e) = \frac{1}{2} (\ln(5) - \ln(8)) \][/tex]
### Step 3: Evaluating the given options
1. [tex]\( x = \ln(8) - \ln(5) \)[/tex]
- This equation simplifies to [tex]\( \ln\left(\frac{8}{5}\right) \)[/tex]. Since [tex]\( \ln(e) \)[/tex] does not match this expression, it is not equivalent.
2. [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]
- This is exactly the expression we derived for [tex]\( 2\ln(e) \)[/tex]. Thus, this equation is not equivalent to the solution for [tex]\( \ln(e) \)[/tex].
3. [tex]\( x = \ln\left(\frac{8}{5}\right) \)[/tex]
- Similar to option 1, this does not match [tex]\( \ln(e) \)[/tex] and hence not equivalent.
4. [tex]\( x = \ln(5) + \ln(8) \)[/tex]
- This equation simplifies to [tex]\( \ln(5 \cdot 8) = \ln(40) \)[/tex], which is not equivalent to the derived solution.
5. [tex]\( x = \ln(5) - \ln(8) \)[/tex]
- Simplifies to [tex]\( \ln\left(\frac{5}{8}\right) \)[/tex]. Since [tex]\( \ln(e) = \frac{1}{2} (\ln(5) - \ln(8)) \)[/tex], this is equivalent.
6. [tex]\( x = \frac{\ln(5)}{\ln(8)} \)[/tex]
- This equation does not match the solution [tex]\( 2\ln(e) \)[/tex], hence it is not equivalent.
### Conclusion
The only correct answer that gives an equivalent solution is:
[tex]\[ x = \ln(5) - \ln(8) \][/tex]
[tex]\[ 8e^2 - 5 = 0 \][/tex]
### Step 1: Solving for [tex]\( e^2 \)[/tex]
[tex]\[ 8e^2 = 5 \][/tex]
[tex]\[ e^2 = \frac{5}{8} \][/tex]
### Step 2: Taking the natural logarithm on both sides
[tex]\[ \ln(e^2) = \ln\left(\frac{5}{8}\right) \][/tex]
Since [tex]\( \ln(e^2) = 2\ln(e) \)[/tex], we get:
[tex]\[ 2 \ln(e) = \ln\left(\frac{5}{8}\right) \][/tex]
[tex]\[ \ln(e) = \frac{1}{2}\ln\left(\frac{5}{8}\right) \][/tex]
Now let's rewrite [tex]\( \ln\left(\frac{5}{8}\right) \)[/tex] using logarithm properties:
[tex]\[ \ln\left(\frac{5}{8}\right) = \ln(5) - \ln(8) \][/tex]
So:
[tex]\[ \ln(e) = \frac{1}{2} (\ln(5) - \ln(8)) \][/tex]
### Step 3: Evaluating the given options
1. [tex]\( x = \ln(8) - \ln(5) \)[/tex]
- This equation simplifies to [tex]\( \ln\left(\frac{8}{5}\right) \)[/tex]. Since [tex]\( \ln(e) \)[/tex] does not match this expression, it is not equivalent.
2. [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]
- This is exactly the expression we derived for [tex]\( 2\ln(e) \)[/tex]. Thus, this equation is not equivalent to the solution for [tex]\( \ln(e) \)[/tex].
3. [tex]\( x = \ln\left(\frac{8}{5}\right) \)[/tex]
- Similar to option 1, this does not match [tex]\( \ln(e) \)[/tex] and hence not equivalent.
4. [tex]\( x = \ln(5) + \ln(8) \)[/tex]
- This equation simplifies to [tex]\( \ln(5 \cdot 8) = \ln(40) \)[/tex], which is not equivalent to the derived solution.
5. [tex]\( x = \ln(5) - \ln(8) \)[/tex]
- Simplifies to [tex]\( \ln\left(\frac{5}{8}\right) \)[/tex]. Since [tex]\( \ln(e) = \frac{1}{2} (\ln(5) - \ln(8)) \)[/tex], this is equivalent.
6. [tex]\( x = \frac{\ln(5)}{\ln(8)} \)[/tex]
- This equation does not match the solution [tex]\( 2\ln(e) \)[/tex], hence it is not equivalent.
### Conclusion
The only correct answer that gives an equivalent solution is:
[tex]\[ x = \ln(5) - \ln(8) \][/tex]
