Question

Find the value of m, if (3/5) raise to −3 multiply 5/3 raise to 11 equals 3/5 raise to 3m+1

Answer 1

`m=-5`

Explanation:

`\left((3)/(5)\right)^(-3)\left((5)/(3)\right)^(11)=\left((3)/(5)\right)^(3m+1)\\\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)\\\ln \left(\left((3)/(5)\right)^(-3)\left((5)/(3)\right)^(11)\right)=\ln \left(\left((3)/(5)\right)^(3m+1)\right)\\\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)`

`\ln \left(\left((3)/(5)\right)^(3m+1)\right)=\left(3m+1\right)\ln \left((3)/(5)\right)\\\ln \left(\left((3)/(5)\right)^(-3)\left((5)/(3)\right)^(11)\right)=\left(3m+1\right)\ln \left((3)/(5)\right)\\\mathrm{Solve\:}\:\ln \left(\left((3)/(5)\right)^(-3)\left((5)/(3)\right)^(11)\right)=\left(3m+1\right)\ln \left((3)/(5)\right):\quad m=(14\ln \left(5\right)-14\ln \left(3\right)-\ln \left((3)/(5)\right))/(3\ln \left((3)/(5)\right))`

`m=(14\ln \left(5\right)-14\ln \left(3\right)-\ln \left((3)/(5)\right))/(3\ln \left((3)/(5)\right))\\\mathrm{Decimal}:\quad m=-5`