Simplify the expression (1-log_((1)/(a))((1)/((a-b)^(2)))+log_(a)^(2)(a-b))/((1-log_(sqrta)(a-b)+log_(a)^(2)(a-b))^((1)/(2))).

Recognize Log Properties: Recognize that log properties can be used to simplify the expression. (1-log_((1)/(a))((1)/((a-b)^(2)))+log_(a)^(2)(a-b))/((1-log_(sqrta)(a-b)+log_(a)^(2)(a-b))^((1)/(2)))Use Change of Base Formula: Use the change of base formula: log_b(a) = log_c(a) / log_c(b). log_((1)/(a))((1)/((a-b)^(2))) = log((1)/((a-b)^(2))) / log((1)/(a))Simplify Logarithm: Simplify the logarithm using the property log(1/x) = -log(x). log((1)/((a-b)^(2))) = -log((a-b)^(2))Apply Power Rule: Apply the power rule of logarithms: log(x^y) = y*log(x). -log((a-b)^(2)) = -2*log(a-b)Substitute Simplified Log: Substitute the simplified log back into the expression. (1 - (-2*log(a-b)) + log_(a)^(2)(a-b)) / ((1 - log_(sqrta)(a-b) + log_(a)^(2)(a-b))^((1)/(2)))Correct Log Notation: Recognize that log_(a)^(2)(x) is not a standard logarithm notation and correct it. log_(a)^(2)(x) should be (log_a(x))^2.

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Simplify the expression $\newline$ $(1-\log_{\left(\frac{1}{a}\right)}\left(\frac{1}{(a-b)^{2}}\right)+\log_{a}^{2}(a-b))/\left((1-\log_{\sqrt{a}}(a-b)+\log_{a}^{2}(a-b))^{\frac{1}{2}}\right).$

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Math Problems

Algebra 1

Multiplication with rational exponents

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Q. Simplify the expression

\newline

(1-\log_{\left(\frac{1}{a}\right)}\left(\frac{1}{(a-b)^{2}}\right)+\log_{a}^{2}(a-b))/\left((1-\log_{\sqrt{a}}(a-b)+\log_{a}^{2}(a-b))^{\frac{1}{2}}\right).

Recognize Log Properties: Recognize that log properties can be used to simplify the expression. $\frac{1-\log_{\frac{1}{a}}\left(\frac{1}{(a-b)^{2}}\right)+\log_{a}^{2}(a-b)}{\left(1-\log_{\sqrt{a}}(a-b)+\log_{a}^{2}(a-b)\right)^{\frac{1}{2}}}$
Use Change of Base Formula: Use the change of base formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ . $\log_{\frac{1}{a}}\left(\frac{1}{(a-b)^{2}}\right) = \frac{\log\left(\frac{1}{(a-b)^{2}}\right)}{\log\left(\frac{1}{a}\right)}$
Simplify Logarithm: Simplify the logarithm using the property $\log\left(\frac{1}{x}\right) = -\log(x)$ . $\log\left(\frac{1}{(a-b)^{2}}\right) = -\log((a-b)^{2})$
Apply Power Rule: Apply the power rule of logarithms: $\log(x^y) = y\log(x)$ . $-\log((a-b)^{2}) = -2\log(a-b)$
Substitute Simplified Log: Substitute the simplified log back into the expression. $\newline$ $(1 - (-2\log(a-b)) + \log_{a}^{2}(a-b)) / ((1 - \log_{\sqrt{a}}(a-b) + \log_{a}^{2}(a-b))^{(1)/(2)})$
Correct Log Notation: Recognize that $\log_{a}^{2}(x)$ is not a standard logarithm notation and correct it. $\log_{a}^{2}(x)$ should be $(\log_{a}(x))^2$ .