Question Number 4515 by love math last updated on 04/Feb/16
$$\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{5}×\mathrm{9}^{\frac{\mathrm{2}}{{x}}} \\ $$
Commented by love math last updated on 05/Feb/16
$${Is}\:{there}\:{any}\:{way}\:{to}\:{prove}\:\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{45}\:{without}\:{using}\:{calculator}? \\ $$$$ \\ $$
Commented by Rasheed Soomro last updated on 06/Feb/16
$$\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{45} \\ $$$${LHS}: \\ $$$$\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{3}^{\mathrm{2}} .\mathrm{3}^{{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{9}×\mathrm{3}^{\frac{{log}\:\mathrm{25}}{{log}\:\mathrm{9}}} \\ $$$$=\mathrm{9}×\mathrm{3}^{\frac{{log}\:\mathrm{5}^{\mathrm{2}} }{{log}\:\mathrm{3}^{\mathrm{2}} }} =\mathrm{9}×\mathrm{3}^{\frac{\mathrm{2}{log}\:\mathrm{5}}{\mathrm{2}{log}\:\mathrm{3}}} =\mathrm{9}×\mathrm{3}^{\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}} \\ $$$$=\mathrm{9}×\mathrm{3}^{{log}_{\mathrm{3}} \mathrm{5}} =\mathrm{9}×\mathrm{5}=\mathrm{45}={RHS} \\ $$$$……… \\ $$$${Let}\:{y}=\mathrm{3}^{{log}_{\mathrm{3}} \mathrm{5}} \\ $$$$\:\:\:\:\:\:\:\:\:{log}_{\mathrm{3}} {y}={log}_{\mathrm{3}} \left(\mathrm{3}^{{log}_{\mathrm{3}} \mathrm{5}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({log}_{\mathrm{3}} \mathrm{5}\right)\:{log}_{\mathrm{3}} \mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({log}_{\mathrm{3}} \mathrm{5}\right)\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={log}_{\mathrm{3}} \mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:{log}_{\mathrm{3}} {y}={log}_{\mathrm{3}} \mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}=\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\:\:\mathrm{3}^{{log}_{\mathrm{3}} \mathrm{5}} =\mathrm{5} \\ $$
Answered by Rasheed Soomro last updated on 05/Feb/16
$$\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{5}×\mathrm{9}^{\frac{\mathrm{2}}{{x}}} \\ $$$${In}\:{order}\:{to}\:{write}\:\mathrm{5}\:{as}\:{power}\:{of}\:\mathrm{3}\::\: \\ $$$${let}\:\mathrm{3}^{{y}} =\mathrm{5} \\ $$$$\:\:\:\:\:{ylog}\:\mathrm{3}={log}\:\mathrm{5} \\ $$$$\:\:\:\:\:{y}=\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}} \\ $$$$\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{3}^{\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}} ×\mathrm{3}^{\frac{\mathrm{4}}{{x}}} \\ $$$$\mathrm{3}^{\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25}} =\mathrm{3}^{\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}+\frac{\mathrm{4}}{{x}}} \\ $$$$\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}+\frac{\mathrm{4}}{{x}}=\mathrm{2}+{log}_{\mathrm{9}} \mathrm{25} \\ $$$$\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}+\frac{\mathrm{4}}{{x}}=\mathrm{2}+\frac{{log}\:\mathrm{25}}{{log}\:\mathrm{9}} \\ $$$$\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}+\frac{\mathrm{4}}{{x}}=\mathrm{2}+\frac{{log}\:\mathrm{5}^{\mathrm{2}} }{{log}\:\mathrm{3}^{\mathrm{2}} } \\ $$$$\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}+\frac{\mathrm{4}}{{x}}=\mathrm{2}+\frac{\mathrm{2}{log}\:\mathrm{5}}{\mathrm{2}{log}\:\mathrm{3}} \\ $$$$\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}}+\frac{\mathrm{4}}{{x}}=\mathrm{2}+\frac{{log}\:\mathrm{5}}{{log}\:\mathrm{3}} \\ $$$$\frac{\mathrm{4}}{{x}}=\mathrm{2} \\ $$$${x}=\mathrm{2} \\ $$