Solve the following system of logarithmic equations:
\(\log(x) + \log(y) = 2\)
\(\log(x) – \log(y) = 1\)
Your task is to:
(a) Set up the equivalent exponential equations using logarithmic properties.
(b) Solve the system for \(x\) and \(y\).
(c) Validate that the solutions satisfy the domain restrictions for the logarithm.
Question
Starting with the system:
\(\log(x) + \log(y) = 2\)
\(\log(x) – \log(y) = 1\)
Use logarithm properties to rewrite each equation in exponential form.
Hint
Hint
Hint
Combine the logarithms using the properties \(\log(a) + \log(b) = \log(ab)\) and \(\log(a) – \log(b) = \log(\frac{a}{b})\). Remember that \(\log(x)\) implies base 10 logarithm.
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Answer
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- Formula Booklet
Complete Answer
The equations become \(xy = 10^2 = 100\) and \(\frac{x}{y} = 10^1 = 10\).
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Question
Solve the system of equations obtained in part (a):
\(xy = 100\)
\(\frac{x}{y} = 10\)
Hint
Hint
Hint
From the second equation, express \(x\) in terms of \(y\) and substitute this expression into the first equation to solve for \(y\).
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Complete Answer
The solution is \(x = 10\sqrt{10}\) and \(y = \sqrt{10}\).
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Question
Validate the solution by ensuring that it meets the domain restrictions for logarithmic functions.
Hint
Hint
Hint
For \(\log(x)\) and \(\log(y)\) to be defined, both \(x\) and \(y\) must be strictly greater than zero. Check if the obtained values of \(x\) and \(y\) satisfy these conditions.
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Complete Answer
Since \(x = 10\sqrt{10} > 0\) and \(y = \sqrt{10} > 0\), the solution is valid.
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