Example Question - single logarithm
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Combining Logarithmic Expressions into a Single Logarithm
<p>The expression given is \( \ln(7) + 4\ln(4) \).</p> <p>First, use the logarithm power rule \( a\ln(b) = \ln(b^a) \) to rewrite \( 4\ln(4) \) as \( \ln(4^4) \).</p> <p>Thus, \( 4\ln(4) = \ln(4^4) = \ln(256) \).</p> <p>Now, use the logarithm addition rule \( \ln(a) + \ln(b) = \ln(a \cdot b) \) to combine \( \ln(7) \) and \( \ln(256) \) into a single logarithm.</p> <p>\( \ln(7) + \ln(256) = \ln(7 \cdot 256) = \ln(1792) \).</p> <p>Therefore, \( \ln(7) + 4\ln(4) \) can be expressed as a single logarithm \( \ln(1792) \).</p>
Solving and Combining Logarithmic Expressions
To solve the expression given, 2 log 5 + 5 log x, and express it as a single logarithm, we will utilize the properties of logarithms: 1. The power rule: log(a^b) = b * log(a) 2. The product rule: log(a) + log(b) = log(a * b) Let's apply these rules step by step: The first term 2 log 5 can be rewritten using the power rule as: log(5^2) = log(25) The second term 5 log x can be rewritten using the power rule as: log(x^5) Now, adding the two log terms using the product rule: log(25) + log(x^5) = log(25 * x^5) Now the expression is written as a single logarithm: log(25 * x^5) Looking at the answer choices, the correct answer is B. log(25 * x^5).
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