Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Certainly! Let's solve the given equation step-by-step and verify which of the given logarithmic expressions are equivalent solutions.
We start with the given equation:
[tex]\[ 8 e^x - 5 = 0 \][/tex]
1. Isolate the exponential term:
[tex]\[ 8 e^x = 5 \][/tex]
2. Solve for [tex]\( e^x \)[/tex]:
[tex]\[ e^x = \frac{5}{8} \][/tex]
3. Take the natural logarithm of both sides:
[tex]\[ \ln(e^x) = \ln\left(\frac{5}{8}\right) \][/tex]
4. Simplify using properties of logarithms:
[tex]\[ x \ln(e) = \ln\left(\frac{5}{8}\right) \][/tex]
Since [tex]\(\ln(e) = 1\)[/tex], we have:
[tex]\[ x = \ln\left(\frac{5}{8}\right) \][/tex]
So we have:
[tex]\[ x = \ln\left(\frac{5}{8}\right) \][/tex]
Next, let's check the given options to see which ones are equivalent to [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]:
1. [tex]\( x = \ln(5) - \ln(8) \)[/tex]:
Using the property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we have:
[tex]\[\ln\left(\frac{5}{8}\right) = \ln(5) - \ln(8)\][/tex]
This is an equivalent expression.
2. [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]:
This is already our derived solution.
This is an equivalent expression.
3. [tex]\( x = \ln(5) + \ln(8) \)[/tex]:
Using the property [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex], we see:
[tex]\[\ln(5) + \ln(8) = \ln(40)\][/tex]
This is not an equivalent expression.
4. [tex]\( x = \frac{\ln(5)}{\ln(8)} \)[/tex]:
This form doesn't represent a valid manipulation of the given equation and has no direct logarithmic property to simplify into the correct form.
This is not an equivalent expression.
5. [tex]\( x = \ln\left(\frac{8}{5}\right) \)[/tex]:
Using the property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we have:
[tex]\[\ln\left(\frac{8}{5}\right) = \ln(8) - \ln(5)\][/tex]
This is not an equivalent expression because it represents the inverse ratio.
6. [tex]\( x = \ln(8) - \ln(5) \)[/tex]:
Using the property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we see:
[tex]\[\ln(8) - \ln(5) = \ln\left(\frac{8}{5}\right)\][/tex]
This is not an equivalent expression.
So, the correct choices are:
- [tex]\( x = \ln(5) - \ln(8) \)[/tex]
- [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]
The correct options are:
[tex]\[ \boxed{1 \text{ and } 2} \][/tex]
We start with the given equation:
[tex]\[ 8 e^x - 5 = 0 \][/tex]
1. Isolate the exponential term:
[tex]\[ 8 e^x = 5 \][/tex]
2. Solve for [tex]\( e^x \)[/tex]:
[tex]\[ e^x = \frac{5}{8} \][/tex]
3. Take the natural logarithm of both sides:
[tex]\[ \ln(e^x) = \ln\left(\frac{5}{8}\right) \][/tex]
4. Simplify using properties of logarithms:
[tex]\[ x \ln(e) = \ln\left(\frac{5}{8}\right) \][/tex]
Since [tex]\(\ln(e) = 1\)[/tex], we have:
[tex]\[ x = \ln\left(\frac{5}{8}\right) \][/tex]
So we have:
[tex]\[ x = \ln\left(\frac{5}{8}\right) \][/tex]
Next, let's check the given options to see which ones are equivalent to [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]:
1. [tex]\( x = \ln(5) - \ln(8) \)[/tex]:
Using the property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we have:
[tex]\[\ln\left(\frac{5}{8}\right) = \ln(5) - \ln(8)\][/tex]
This is an equivalent expression.
2. [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]:
This is already our derived solution.
This is an equivalent expression.
3. [tex]\( x = \ln(5) + \ln(8) \)[/tex]:
Using the property [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex], we see:
[tex]\[\ln(5) + \ln(8) = \ln(40)\][/tex]
This is not an equivalent expression.
4. [tex]\( x = \frac{\ln(5)}{\ln(8)} \)[/tex]:
This form doesn't represent a valid manipulation of the given equation and has no direct logarithmic property to simplify into the correct form.
This is not an equivalent expression.
5. [tex]\( x = \ln\left(\frac{8}{5}\right) \)[/tex]:
Using the property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we have:
[tex]\[\ln\left(\frac{8}{5}\right) = \ln(8) - \ln(5)\][/tex]
This is not an equivalent expression because it represents the inverse ratio.
6. [tex]\( x = \ln(8) - \ln(5) \)[/tex]:
Using the property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we see:
[tex]\[\ln(8) - \ln(5) = \ln\left(\frac{8}{5}\right)\][/tex]
This is not an equivalent expression.
So, the correct choices are:
- [tex]\( x = \ln(5) - \ln(8) \)[/tex]
- [tex]\( x = \ln\left(\frac{5}{8}\right) \)[/tex]
The correct options are:
[tex]\[ \boxed{1 \text{ and } 2} \][/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.