Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Ask your questions and receive precise answers from experienced professionals across different disciplines. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To solve the equation [tex]\( 2^{5x - 1} = 3^x \)[/tex], follow these steps:
1. Understand the nature of the equation:
We have an equation involving exponential functions with different bases, [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
2. Transform the equation using logarithms:
To handle the exponents with different bases, we apply logarithms (natural logarithms, for simplicity):
[tex]\[ \ln(2^{5x - 1}) = \ln(3^x) \][/tex]
3. Use logarithmic properties to simplify:
Apply the property of logarithms that allows you to bring down the exponent:
[tex]\[ (5x - 1) \ln(2) = x \ln(3) \][/tex]
4. Rearrange to isolate [tex]\(x\)[/tex]:
Let's distribute the logarithm terms and solve for [tex]\(x\)[/tex]:
[tex]\[ 5x \ln(2) - \ln(2) = x \ln(3) \][/tex]
[tex]\[ 5x \ln(2) - x \ln(3) = \ln(2) \][/tex]
5. Factor [tex]\(x\)[/tex] from the left-hand side:
[tex]\[ x (5 \ln(2) - \ln(3)) = \ln(2) \][/tex]
6. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by [tex]\((5 \ln(2) - \ln(3))\)[/tex]:
[tex]\[ x = \frac{\ln(2)}{5 \ln(2) - \ln(3)} \][/tex]
7. Express in a simpler form:
Recognize that the solution can be expressed in terms of another logarithm. Here, we simplify:
[tex]\[ x = \log_{32/3}(2) \][/tex]
Note that [tex]\(\log_{32/3}(2)\)[/tex] can be interpreted using the change of base formula, but it effectively captures the solution in a compact form.
The precise solution is:
[tex]\[ x = \log_{32/3}(2) \][/tex]
1. Understand the nature of the equation:
We have an equation involving exponential functions with different bases, [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
2. Transform the equation using logarithms:
To handle the exponents with different bases, we apply logarithms (natural logarithms, for simplicity):
[tex]\[ \ln(2^{5x - 1}) = \ln(3^x) \][/tex]
3. Use logarithmic properties to simplify:
Apply the property of logarithms that allows you to bring down the exponent:
[tex]\[ (5x - 1) \ln(2) = x \ln(3) \][/tex]
4. Rearrange to isolate [tex]\(x\)[/tex]:
Let's distribute the logarithm terms and solve for [tex]\(x\)[/tex]:
[tex]\[ 5x \ln(2) - \ln(2) = x \ln(3) \][/tex]
[tex]\[ 5x \ln(2) - x \ln(3) = \ln(2) \][/tex]
5. Factor [tex]\(x\)[/tex] from the left-hand side:
[tex]\[ x (5 \ln(2) - \ln(3)) = \ln(2) \][/tex]
6. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by [tex]\((5 \ln(2) - \ln(3))\)[/tex]:
[tex]\[ x = \frac{\ln(2)}{5 \ln(2) - \ln(3)} \][/tex]
7. Express in a simpler form:
Recognize that the solution can be expressed in terms of another logarithm. Here, we simplify:
[tex]\[ x = \log_{32/3}(2) \][/tex]
Note that [tex]\(\log_{32/3}(2)\)[/tex] can be interpreted using the change of base formula, but it effectively captures the solution in a compact form.
The precise solution is:
[tex]\[ x = \log_{32/3}(2) \][/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.