log_(11)((x)/(sqrtx))+log_(11)(sqrt(x^(5)))-(7)/(3)log_(11)x

Simplify logarithm expression: First, simplify log_(11)((x)/(sqrtx)). Using the quotient rule of logarithms, log_b(a/c) = log_b(a) - log_b(c), we get: log_(11)(x) - log_(11)(sqrt(x)). Since sqrt(x) = x^(1/2), rewrite the expression: log_(11)(x) - log_(11)(x^(1/2)). Using the power rule, log_b(a^c) = c*log_b(a), we have: log_(11)(x) - (1/2)*log_(11)(x). Simplify further: (1 - 1/2) * log_(11)(x) = (1/2) * log_(11)(x).Rewrite and simplify: Next, simplify log_(11)(sqrt(x^(5))). Rewrite sqrt(x^(5)) as (x^(5))^(1/2) = x^(5/2). Using the power rule: log_(11)(x^(5/2)) = (5/2) * log_(11)(x).Combine results and simplify: Combine the results from the previous steps: (1/2) * log_(11)(x) + (5/2) * log_(11)(x) - (7/3) * log_(11)(x). Combine like terms: (1/2 + 5/2 - 7/3) * log_(11)(x). Convert fractions to a common denominator and simplify: (3/6 + 15/6 - 14/6) * log_(11)(x) = (4/6) * log_(11)(x). Simplify the fraction: (2/3) * log_(11)(x).

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log_(11)((x)/(sqrtx))+log_(11)(sqrt(x^(5)))-(7)/(3)log_(11)x

$\log _{11}\left(\frac{x}{\sqrt{x}}\right)+\log _{11}\left(\sqrt{x^{5}}\right)-\frac{7}{3} \log _{11} x$

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Algebra 1

Multiplication with rational exponents

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\log _{11}\left(\frac{x}{\sqrt{x}}\right)+\log _{11}\left(\sqrt{x^{5}}\right)-\frac{7}{3} \log _{11} x

Simplify logarithm expression: First, simplify $\log_{11}\left(\frac{x}{\sqrt{x}}\right)$ . Using the quotient rule of logarithms, $\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)$ , we get: $\log_{11}(x) - \log_{11}(\sqrt{x})$ . Since $\sqrt{x} = x^{1/2}$ , rewrite the expression: $\log_{11}(x) - \log_{11}(x^{1/2})$ . Using the power rule, $\log_b(a^c) = c\cdot\log_b(a)$ , we have: $\log_{11}(x) - \left(\frac{1}{2}\right)\cdot\log_{11}(x)$ . Simplify further: $\left(1 - \frac{1}{2}\right) \cdot \log_{11}(x) = \left(\frac{1}{2}\right) \cdot \log_{11}(x)$ .
Rewrite and simplify: Next, simplify $\log_{11}(\sqrt{x^{5}})$ . Rewrite $\sqrt{x^{5}}$ as $(x^{5})^{\frac{1}{2}} = x^{\frac{5}{2}}$ . Using the power rule: $\log_{11}(x^{\frac{5}{2}}) = \frac{5}{2} \cdot \log_{11}(x)$ .
Combine results and simplify: Combine the results from the previous steps: $\newline$ $(\frac{1}{2}) \cdot \log_{11}(x) + (\frac{5}{2}) \cdot \log_{11}(x) - (\frac{7}{3}) \cdot \log_{11}(x)$ . $\newline$ Combine like terms: $\newline$ $(\frac{1}{2} + \frac{5}{2} - \frac{7}{3}) \cdot \log_{11}(x)$ . $\newline$ Convert fractions to a common denominator and simplify: $\newline$ $(\frac{3}{6} + \frac{15}{6} - \frac{14}{6}) \cdot \log_{11}(x) = (\frac{4}{6}) \cdot \log_{11}(x)$ . $\newline$ Simplify the fraction: $\newline$ $(\frac{2}{3}) \cdot \log_{11}(x)$ .