Example Question - solving logarithmic expressions
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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).
Solving Logarithmic Expressions with Properties
To solve the expression given in the image as a single logarithm, we can use the properties of logarithms: The original expression given is: \[ \frac{1}{4} \ln 4 - 2 \ln x \] Firstly, we can use the power rule of logarithms which states that \( \log_b (a^n) = n \log_b a \). We can apply this rule on the first term to bring down the 1/4 as a power of 4 inside the logarithm: \[ = \ln 4^{1/4} - 2 \ln x \] \[ = \ln 2 - 2 \ln x \] (Note: \( 4^{1/4} \) is the fourth root of 4, which is equal to 2.) Next, we use another property of logarithms: the coefficient in front of a logarithm can be converted into a power inside the logarithm, which is: \[ \log_b a^c = c \log_b a \] Applying this to the second term: \[ = \ln 2 - \ln x^2 \] Now, we use the subtraction rule for logarithms, which states: \( \log_b a - \log_b c = \log_b \frac{a}{c} \). We'll apply this to the expression: \[ = \ln \frac{2}{x^2} \] Looking at the options provided in the image, the one that matches is B: \[ \ln \left( \frac{2}{x^2} \right) \]
Solving Logarithmic Expressions with Log Rules
To solve the expression given in the image, \( \log_{c^2}(\frac{a^5b}{c^2}) \), we can apply log rules (quotient rule, power rule, and change of base formula). The quotient rule states that \( \log(\frac{x}{y}) = \log(x) - \log(y) \). The power rule states that \( \log(x^a) = a \log(x) \). The change of base formula states that \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). Let's break down the expression by using these rules: 1. Apply the quotient rule to the argument of the log. \( \log_{c^2}(a^5b) - \log_{c^2}(c^2) \) 2. Apply the power rule to \( a^5 \) and \( c^2 \) within the log expressions. \( 5 \log_{c^2}(a) + \log_{c^2}(b) - 2 \log_{c^2}(c) \) 3. Notice that \( \log_{c^2}(c^2) = 1 \) because the log base and the argument are the same value raised to the same power. Hence, the expression becomes: \( 5 \log_{c^2}(a) + \log_{c^2}(b) - 2 \) 4. Apply the change of base formula to the two remaining log terms. Since we want to express everything in terms of the base c, we get: \( 5 \frac{\log_{c}(a)}{\log_{c}(c^2)} + \frac{\log_{c}(b)}{\log_{c}(c^2)} - 2 \) 5. Simplify by recognizing that \( \log_{c}(c^2) = 2 \). \( 5 \frac{\log_{c}(a)}{2} + \frac{\log_{c}(b)}{2} - 2 \) 6. Simplify the fractions. \( \frac{5}{2} \log_{c}(a) + \frac{1}{2} \log_{c}(b) - 2 \) 7. Looking at the answer choices given, we can see that our derived expression matches choice D when we distribute the 2 outside of the log: \( \frac{5}{2} \log_{c}(a) + \frac{1}{2} \log_{c}(b) - 2 \cdot 1 \) \( \frac{5}{2} \log_{c}(a) + \frac{1}{2} \log_{c}(b) - 2 \log_{c}(c) \) Since \( \log_{c}(c) = 1 \), the 2 just becomes \( 2 \log_{c}(c) \), and thus the answer is: \( \frac{5}{2} \cdot \log_{c}(a) + \frac{1}{2} \cdot \log_{c}(b) - 2 \log_{c}(c) \) So, the correct answer is D: \( 5 \log_{c}(a) + \log_{c}(b) - 2 \log_{c}(c) \)
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