Prove the following:
(log x)2−(log y)2=log xy.log xy(\text{log} \space x)^2 - (\text{log} \space y)^2 = \text{log} \space \dfrac{x}{y}.\text{log} \space xy(log x)2−(log y)2=log yx.log xy
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Given,
Simplifying L.H.S. of above equation we get,
⇒(log x)2−(log y)2⇒(log x−log y)(log x+log y)⇒log xy.log xy\Rightarrow (\text{log} \space x)^2 - (\text{log} \space y)^2 \\[1em] \Rightarrow (\text{log} \space x - \text{log} \space y)(\text{log} \space x + \text{log} \space y) \\[1em] \Rightarrow \text{log} \space\dfrac{x}{y}.\text{log} \space xy⇒(log x)2−(log y)2⇒(log x−log y)(log x+log y)⇒log yx.log xy
Since, L.H.S. = R.H.S.,
Hence, proved that (log x)2−(log y)2=log xy.log xy(\text{log} \space x)^2 - (\text{log} \space y)^2 = \text{log} \space \dfrac{x}{y}.\text{log} \space xy(log x)2−(log y)2=log yx.log xy
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Expand loga x7y8÷z43\text{log}_a \space {\sqrt[3]{x^7y^8 ÷ \sqrt[4]{z}}}loga 3x7y8÷4z
Find the value of log3 33−log5 (0.04).\text{log}{\sqrt{3}} \space 3\sqrt{3} - \text{log}5 \space (0.04).log3 33−log5 (0.04).
2log 1113+log 13077−log 5591=log 2.2\text{log} \space \dfrac{11}{13} + \text{log} \space \dfrac{130}{77} - \text{log} \space \dfrac{55}{91} = \text{log} \space 2.2log 1311+log 77130−log 9155=log 2.
If log (m + n) = log m + log n, show that n = mm−1.\dfrac{m}{m - 1}.m−1m.
