Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Sure, let's solve the equation [tex]\(\log_7(3x^3 + x) - \log_7(x) = 2\)[/tex] step-by-step and identify any extraneous solutions.
First, let's use the properties of logarithms to combine the logarithms:
[tex]\[ \log_7(3x^3 + x) - \log_7(x) = \log_7\left(\frac{3x^3 + x}{x}\right) = \log_7(3x^2 + 1) \][/tex]
So, the equation simplifies to:
[tex]\[ \log_7(3x^2 + 1) = 2 \][/tex]
To clear the logarithm, rewrite the equation in exponential form:
[tex]\[ 3x^2 + 1 = 7^2 \][/tex]
Since [tex]\(7^2 = 49\)[/tex], the equation becomes:
[tex]\[ 3x^2 + 1 = 49 \][/tex]
Now, solve for [tex]\(x\)[/tex]:
[tex]\[ 3x^2 + 1 = 49 \][/tex]
[tex]\[ 3x^2 = 48 \][/tex]
[tex]\[ x^2 = 16 \][/tex]
[tex]\[ x = \pm 4 \][/tex]
So, the potential solutions are [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex].
Next, we should check for extraneous solutions. A solution is extraneous if it does not satisfy the original equation or if it makes the argument of any logarithm non-positive (since logarithms of non-positive numbers are undefined).
Let's check both solutions:
1. For [tex]\(x = 4\)[/tex]:
[tex]\[ \log_7(3(4)^3 + 4) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7(3 \cdot 64 + 4) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7(192 + 4) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7(196) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7\left(\frac{196}{4}\right) = 2 \][/tex]
[tex]\[ \log_7(49) = 2 \][/tex]
Since [tex]\(7^2 = 49\)[/tex], this is true, so [tex]\(x = 4\)[/tex] is a valid solution.
2. For [tex]\(x = -4\)[/tex]:
Check the arguments of the logarithms:
[tex]\[ \log_7(3(-4)^3 + (-4)) - \log_7(-4) \][/tex]
[tex]\[ 3(-64) + (-4) = -192 - 4 = -196 \][/tex]
Since [tex]\(\log_7(-196)\)[/tex] and [tex]\(\log_7(-4)\)[/tex] are undefined (logarithms of negative numbers are undefined in the real number domain), [tex]\(x = -4\)[/tex] is not a valid solution. Therefore, [tex]\(x = -4\)[/tex] is extraneous.
Considering this, the extraneous solution to the given logarithmic equation is:
[tex]\[ x = -4 \][/tex]
First, let's use the properties of logarithms to combine the logarithms:
[tex]\[ \log_7(3x^3 + x) - \log_7(x) = \log_7\left(\frac{3x^3 + x}{x}\right) = \log_7(3x^2 + 1) \][/tex]
So, the equation simplifies to:
[tex]\[ \log_7(3x^2 + 1) = 2 \][/tex]
To clear the logarithm, rewrite the equation in exponential form:
[tex]\[ 3x^2 + 1 = 7^2 \][/tex]
Since [tex]\(7^2 = 49\)[/tex], the equation becomes:
[tex]\[ 3x^2 + 1 = 49 \][/tex]
Now, solve for [tex]\(x\)[/tex]:
[tex]\[ 3x^2 + 1 = 49 \][/tex]
[tex]\[ 3x^2 = 48 \][/tex]
[tex]\[ x^2 = 16 \][/tex]
[tex]\[ x = \pm 4 \][/tex]
So, the potential solutions are [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex].
Next, we should check for extraneous solutions. A solution is extraneous if it does not satisfy the original equation or if it makes the argument of any logarithm non-positive (since logarithms of non-positive numbers are undefined).
Let's check both solutions:
1. For [tex]\(x = 4\)[/tex]:
[tex]\[ \log_7(3(4)^3 + 4) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7(3 \cdot 64 + 4) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7(192 + 4) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7(196) - \log_7(4) = 2 \][/tex]
[tex]\[ \log_7\left(\frac{196}{4}\right) = 2 \][/tex]
[tex]\[ \log_7(49) = 2 \][/tex]
Since [tex]\(7^2 = 49\)[/tex], this is true, so [tex]\(x = 4\)[/tex] is a valid solution.
2. For [tex]\(x = -4\)[/tex]:
Check the arguments of the logarithms:
[tex]\[ \log_7(3(-4)^3 + (-4)) - \log_7(-4) \][/tex]
[tex]\[ 3(-64) + (-4) = -192 - 4 = -196 \][/tex]
Since [tex]\(\log_7(-196)\)[/tex] and [tex]\(\log_7(-4)\)[/tex] are undefined (logarithms of negative numbers are undefined in the real number domain), [tex]\(x = -4\)[/tex] is not a valid solution. Therefore, [tex]\(x = -4\)[/tex] is extraneous.
Considering this, the extraneous solution to the given logarithmic equation is:
[tex]\[ x = -4 \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.