Logarithmic Equations

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What is a logarithmic equation?
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A logarithmic equation is an equation that involves a logarithm of an expression set equal to a number or another expression.
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How do you solve a basic logarithmic equation, such as \( \log_b(x) = c \)?
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To solve \( \log_b(x) = c \), rewrite the equation in exponential form as \( x = b^c \).
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What must be true about the base and argument in a logarithmic equation?
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The base must be positive and not equal to 1, and the argument must be positive.
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How can properties of logarithms simplify solving logarithmic equations?
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Properties such as the product, quotient, and power rules can simplify expressions and isolate the variable.
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What is the product rule for logarithms?
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The product rule states \( \log_b(MN) = \log_b(M) + \log_b(N) \).
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What is the power rule for logarithms?
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The power rule states \( \log_b(M^n) = n \cdot \log_b(M) \).
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What is the quotient rule for logarithms?
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The quotient rule states \( \log_b(M/N) = \log_b(M) - \log_b(N) \).
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How do you solve \( \log_b(x^2) = \log_b(9) \)?
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By the one-to-one property, set \( x^2 = 9 \), solve for \( x \): \( x = 3 \) or \( x = -3 \) (considering restrictions).
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Why can extraneous solutions occur in logarithmic equations?
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Extraneous solutions occur due to taking roots and combining logs that might not hold under original restrictions.
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What is the change of base formula for logarithms?
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The change of base formula is \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), for any positive k.
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How does the change of base formula help solve equations?
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It allows rewriting logarithms in terms of a different base, often converting them to a base easier to work with, like 10 or e.
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How can you verify a solution to a logarithmic equation?
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Substitute the solution back into the original equation and check if both sides are equal.
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What strategy can be used to solve equations with multiple logarithms?
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Use properties of logs to combine into a single logarithm, then rewrite in exponential form to solve.
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What role does graphing play in solving logarithmic equations?
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Graphing can help visualize solutions and verify answers or identify extraneous solutions.
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What is an example of an equation involving a natural logarithm, and how to solve it?
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For \( \ln(x) = 2 \), rewrite it as \( x = e^2 \).
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