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Question Number 195428 by mathlove last updated on 02/Aug/23

$$3^a=7^b=441$$$$\frac{ab}{a+b}=?$$

Answered by Frix last updated on 02/Aug/23

$$441=3^2{7}^2$$$$a=2(1+\frac{\ln 7}{\ln 3})=2(1+r)$$$$b=2(1+\frac{\ln 3}{\ln 7})=2(1+r^{-1})$$$$\frac{ab}{a+b}=\frac{\frac{4{(r+1)}^2}{r}}{\frac{2{(r+1)}^2}{r}}=2$$

Answered by sandy_delta last updated on 02/Aug/23

$$3^{\mathrm{a}}\times 7^{\mathrm{b}}={441}^2$$$$3^{\mathrm{a}}\times 7^{\mathrm{b}}={(3^2.7^2)}^2$$$$3^{\mathrm{a}}\times 7^{\mathrm{b}}=3^4.7^4$$$$\mathrm{so}\mathrm{a}=4\mathrm{and}\mathrm{b}=4$$$$\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}=\frac{4.4}{4+4}=2$$

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