Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To solve the equation [tex]\( 2 \log_5(3x - 2) - \log_5(x) = 2 \)[/tex], follow these steps:
1. Express the equation in terms of logarithms:
We start with the given equation:
[tex]\[ 2 \log_5(3x - 2) - \log_5(x) = 2 \][/tex]
2. Combine the logarithmic terms:
Recall the properties of logarithms:
[tex]\[ a \log_b(c) = \log_b(c^a) \][/tex]
and
[tex]\[ \log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right) \][/tex]
So we use the first property to rewrite [tex]\(2 \log_5(3x - 2)\)[/tex] as:
[tex]\[ 2 \log_5(3x - 2) = \log_5((3x - 2)^2) \][/tex]
Then our equation becomes:
[tex]\[ \log_5((3x - 2)^2) - \log_5(x) = 2 \][/tex]
Using the second property of logarithms, combine the logarithmic terms:
[tex]\[ \log_5\left(\frac{(3x - 2)^2}{x}\right) = 2 \][/tex]
3. Rewrite the equation in exponential form:
The equation [tex]\(\log_b(A) = C\)[/tex] is equivalent to [tex]\(A = b^C\)[/tex].
Thus, we can rewrite [tex]\(\log_5\left(\frac{(3x - 2)^2}{x}\right) = 2\)[/tex] as:
[tex]\[ \frac{(3x - 2)^2}{x} = 5^2 \][/tex]
Simplify this:
[tex]\[ \frac{(3x - 2)^2}{x} = 25 \][/tex]
4. Solve the resulting equation:
Multiply both sides by [tex]\(x\)[/tex] to eliminate the fraction:
[tex]\[ (3x - 2)^2 = 25x \][/tex]
Expand the left-hand side:
[tex]\[ 9x^2 - 12x + 4 = 25x \][/tex]
Bring all terms to one side to set the equation to zero:
[tex]\[ 9x^2 - 12x + 4 - 25x = 0 \][/tex]
[tex]\[ 9x^2 - 37x + 4 = 0 \][/tex]
5. Solve the quadratic equation:
Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 9\)[/tex], [tex]\(b = -37\)[/tex], and [tex]\(c = 4\)[/tex]:
[tex]\[ x = \frac{-(-37) \pm \sqrt{(-37)^2 - 4 \cdot 9 \cdot 4}}{2 \cdot 9} \][/tex]
[tex]\[ x = \frac{37 \pm \sqrt{1369 - 144}}{18} \][/tex]
[tex]\[ x = \frac{37 \pm \sqrt{1225}}{18} \][/tex]
[tex]\[ x = \frac{37 \pm 35}{18} \][/tex]
This gives us two possible solutions:
[tex]\[ x = \frac{37 + 35}{18} = \frac{72}{18} = 4 \][/tex]
[tex]\[ x = \frac{37 - 35}{18} = \frac{2}{18} = \frac{1}{9} \][/tex]
6. Verify the solutions:
Substitute [tex]\(x = 4\)[/tex] and [tex]\(x = \frac{1}{9}\)[/tex] back into the original equation to check for validity:
- For [tex]\(x = 4\)[/tex]:
[tex]\[ 2 \log_5(3 \cdot 4 - 2) - \log_5(4) = 2 \][/tex]
[tex]\[ 2 \log_5(10) - \log_5(4) = 2 \][/tex]
[tex]\[ 2 \log_5(10) - \log_5(4) = 2 \][/tex]
Simplifying the left-hand side confirms that the equation holds true.
- For [tex]\(x = \frac{1}{9}\)[/tex]:
Without calculation, substitute [tex]\(x = \frac{1}{9}\)[/tex] into the equation results negative arguments in the logarithms which are undefined in the real number ([3x - 2 < 0]). Therefore, this solution doesn't satisfy the original equation,
Thus, the valid solution is:
[tex]\[ x = 4 \][/tex]
1. Express the equation in terms of logarithms:
We start with the given equation:
[tex]\[ 2 \log_5(3x - 2) - \log_5(x) = 2 \][/tex]
2. Combine the logarithmic terms:
Recall the properties of logarithms:
[tex]\[ a \log_b(c) = \log_b(c^a) \][/tex]
and
[tex]\[ \log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right) \][/tex]
So we use the first property to rewrite [tex]\(2 \log_5(3x - 2)\)[/tex] as:
[tex]\[ 2 \log_5(3x - 2) = \log_5((3x - 2)^2) \][/tex]
Then our equation becomes:
[tex]\[ \log_5((3x - 2)^2) - \log_5(x) = 2 \][/tex]
Using the second property of logarithms, combine the logarithmic terms:
[tex]\[ \log_5\left(\frac{(3x - 2)^2}{x}\right) = 2 \][/tex]
3. Rewrite the equation in exponential form:
The equation [tex]\(\log_b(A) = C\)[/tex] is equivalent to [tex]\(A = b^C\)[/tex].
Thus, we can rewrite [tex]\(\log_5\left(\frac{(3x - 2)^2}{x}\right) = 2\)[/tex] as:
[tex]\[ \frac{(3x - 2)^2}{x} = 5^2 \][/tex]
Simplify this:
[tex]\[ \frac{(3x - 2)^2}{x} = 25 \][/tex]
4. Solve the resulting equation:
Multiply both sides by [tex]\(x\)[/tex] to eliminate the fraction:
[tex]\[ (3x - 2)^2 = 25x \][/tex]
Expand the left-hand side:
[tex]\[ 9x^2 - 12x + 4 = 25x \][/tex]
Bring all terms to one side to set the equation to zero:
[tex]\[ 9x^2 - 12x + 4 - 25x = 0 \][/tex]
[tex]\[ 9x^2 - 37x + 4 = 0 \][/tex]
5. Solve the quadratic equation:
Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 9\)[/tex], [tex]\(b = -37\)[/tex], and [tex]\(c = 4\)[/tex]:
[tex]\[ x = \frac{-(-37) \pm \sqrt{(-37)^2 - 4 \cdot 9 \cdot 4}}{2 \cdot 9} \][/tex]
[tex]\[ x = \frac{37 \pm \sqrt{1369 - 144}}{18} \][/tex]
[tex]\[ x = \frac{37 \pm \sqrt{1225}}{18} \][/tex]
[tex]\[ x = \frac{37 \pm 35}{18} \][/tex]
This gives us two possible solutions:
[tex]\[ x = \frac{37 + 35}{18} = \frac{72}{18} = 4 \][/tex]
[tex]\[ x = \frac{37 - 35}{18} = \frac{2}{18} = \frac{1}{9} \][/tex]
6. Verify the solutions:
Substitute [tex]\(x = 4\)[/tex] and [tex]\(x = \frac{1}{9}\)[/tex] back into the original equation to check for validity:
- For [tex]\(x = 4\)[/tex]:
[tex]\[ 2 \log_5(3 \cdot 4 - 2) - \log_5(4) = 2 \][/tex]
[tex]\[ 2 \log_5(10) - \log_5(4) = 2 \][/tex]
[tex]\[ 2 \log_5(10) - \log_5(4) = 2 \][/tex]
Simplifying the left-hand side confirms that the equation holds true.
- For [tex]\(x = \frac{1}{9}\)[/tex]:
Without calculation, substitute [tex]\(x = \frac{1}{9}\)[/tex] into the equation results negative arguments in the logarithms which are undefined in the real number ([3x - 2 < 0]). Therefore, this solution doesn't satisfy the original equation,
Thus, the valid solution is:
[tex]\[ x = 4 \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
