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Question Number 202172 by MATHEMATICSAM last updated on 22/Dec/23

$$\mathrm{If}{\log }_{12}18=\mathrm{A}\mathrm{and}{\log }_{24}54=\mathrm{B}\mathrm{then}\mathrm{prove}$$$$\mathrm{that}\mathrm{AB}+5(\mathrm{A}-\mathrm{B})=1.$$

Answered by Rasheed.Sindhi last updated on 22/Dec/23

$${12}^{\mathrm{A}}=18\mathrm{\&}{24}^{\mathrm{B}}=54$$$${(2^2.3)}^{\mathrm{A}}=18\mathrm{\&}{(2^3.3)}^{\mathrm{B}}=54$$$$2^{2\mathrm{A}}.3^{\mathrm{A}}=2.3^2\mathrm{\&}{2}^{3\mathrm{B}}.3^{\mathrm{B}}=2.3^3$$$$\frac{2^{2\mathrm{A}}.3^{\mathrm{A}}}{2^{3\mathrm{B}}.3^{\mathrm{B}}}=\frac{2.3^2}{2.3^3}=3^{-1}$$$$2^{2\mathrm{A}-3\mathrm{B}}.3^{\mathrm{A}-\mathrm{B}}=2^0.3^{-1}$$$$2\mathrm{A}-3\mathrm{B}=0\wedge \mathrm{A}-\mathrm{B}=-1(\Rightarrow \mathrm{A}=\mathrm{B}-1)$$$$2(\mathrm{B}-1)-3\mathrm{B}=0$$$$\mathrm{B}=-2\Rightarrow \mathrm{A}=-2-1=-3$$$$$$$$\mathrm{AB}+5(\mathrm{A}-\mathrm{B})=1$$$$\Rightarrow \mathrm{AB}+5\mathrm{A}-5\mathrm{B}-25+25=1$$$$\mathrm{A}(\mathrm{B}+5)-5(\mathrm{B}+5)+25=1$$$$(\mathrm{A}-5)(\mathrm{B}+5)+25=1$$$$(-3-5)(-2+5)+25=1$$$$(-8)\left(3\right)+25=1$$$$1=1$$$$\mathrm{QED}$$

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