Write the expression as a single logarithm. 7log_(c)x-(1)/(4)log_(c)w+2log_(c)z

Apply Power Rule: We are given the expression 7log_(c)x - (1/4)log_(c)w + 2log_(c)z. To combine these logarithms into a single logarithm, we will use the logarithm properties: the power rule, which states that a*log_b(x) = log_b(x^a), and the product and quotient rules, which state that log_b(x) + log_b(y) = log_b(xy) and log_b(x) - log_b(y) = log_b(x/y), respectively.Rewrite Using Rules: First, apply the power rule to each term to move the coefficients inside the logarithms as exponents of the arguments. 7log_(c)x becomes log_(c)(x^7) (1/4)log_(c)w becomes log_(c)(w^(1/4)) 2log_(c)z becomes log_(c)(z^2)Combine Terms: Now, rewrite the expression using the product and quotient rules. log_(c)(x^7) - log_(c)(w^(1/4)) + log_(c)(z^2) According to the product rule, we can combine the terms with addition into a single logarithm with a product inside. log_(c)(x^7 * z^2) And according to the quotient rule, we can combine the term with subtraction into a single logarithm with a quotient inside. log_(c)((x^7 * z^2) / w^(1/4))Final Answer: The expression is now simplified to a single logarithm: log_(c)((x^7 * z^2) / w^(1/4)) This is the final answer.

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Write the expression as a single logarithm.

7log_(c)x-(1)/(4)log_(c)w+2log_(c)z

Write the expression as a single logarithm. $\newline$ $7 \log _{c} x-\frac{1}{4} \log _{c} w+2 \log _{c} z$

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Algebra 1

Multiplication with rational exponents

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Q. Write the expression as a single logarithm.

\newline

7 \log _{c} x-\frac{1}{4} \log _{c} w+2 \log _{c} z

Apply Power Rule: We are given the expression $7\log_{c}x - \frac{1}{4}\log_{c}w + 2\log_{c}z$ . To combine these logarithms into a single logarithm, we will use the logarithm properties: the power rule, which states that $a\log_{b}(x) = \log_{b}(x^{a})$ , and the product and quotient rules, which state that $\log_{b}(x) + \log_{b}(y) = \log_{b}(xy)$ and $\log_{b}(x) - \log_{b}(y) = \log_{b}(x/y)$ , respectively.
Rewrite Using Rules: First, apply the power rule to each term to move the coefficients inside the logarithms as exponents of the arguments. $\newline$ $7\log_{c}x$ becomes $\log_{c}(x^7)$ $\newline$ $(1/4)\log_{c}w$ becomes $\log_{c}(w^{1/4})$ $\newline$ $2\log_{c}z$ becomes $\log_{c}(z^2)$
Combine Terms: Now, rewrite the expression using the product and quotient rules. $\newline$ $\log_{c}(x^7) - \log_{c}(w^{\frac{1}{4}}) + \log_{c}(z^2)$ $\newline$ According to the product rule, we can combine the terms with addition into a single logarithm with a product inside. $\newline$ $\log_{c}(x^7 \cdot z^2)$ $\newline$ And according to the quotient rule, we can combine the term with subtraction into a single logarithm with a quotient inside. $\newline$ $\log_{c}\left(\frac{x^7 \cdot z^2}{w^{\frac{1}{4}}}\right)$
Final Answer: The expression is now simplified to a single logarithm: $\log_{c}\left(\frac{x^{7} \cdot z^{2}}{w^{\frac{1}{4}}}\right)$ This is the final answer.