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Question Number 99050 by bemath last updated on 18/Jun/20

$$5^{5-3x}+2^{x+5}=5^{7-3x}-2^{x+6}$$

Commented by bobhans last updated on 18/Jun/20

$$\frac{5^5}{5^{3\mathrm{x}}}+32.2^{\mathrm{x}}=\frac{5^7}{5^{3\mathrm{x}}}-64.2^{\mathrm{x}}$$$$96.2^{\mathrm{x}}=\frac{5^5(5^2-1)}{5^{3\mathrm{x}}}=\frac{24.5^5}{5^{3\mathrm{x}}}$$$$4.2^{\mathrm{x}}=5^{5-3\mathrm{x}}\Rightarrow 2^{2+\mathrm{x}}=5^{5-3\mathrm{x}}$$$$(2+\mathrm{x})\ln \left(2\right)=(5-3\mathrm{x})\ln \left(5\right)$$$$\mathrm{x}\ln \left(2\right)+3\mathrm{x}\ln \left(5\right)=5\ln \left(5\right)-2\ln \left(2\right)$$$$\mathrm{x}(0.693+4.828)=8.047-1.386$$$$\mathrm{x}=\frac{6.661}{5.521}\approx 1.206$$

Commented by bemath last updated on 18/Jun/20

$$\mathrm{thank}\mathrm{you}\mathrm{both}\mathrm{sir}$$$$$$

Answered by Rasheed.Sindhi last updated on 18/Jun/20

$$5^{5-3x}+2^{x+5}=5^{7-3x}-2^{x+6}$$$$2^{x+5}+2^{x+6}=5^{7-3x}-5^{5-3x}$$$$2^{x+5}(1+2)=5^{5-3x}(5^2-1)$$$$\frac{2^{x+5}\times 5^{3x}}{5^5}=\frac{24}{3}$$$$2^5.2^x.{125}^x=\frac{24.5^5}{3}$$$${\left(250\right)}^x=\frac{24.5^5}{3.2^5}$$$${\left(250\right)}^x=\frac{8.5^5}{2^5}=\frac{5^5}{2^2}$$$$x=\frac{5\log 5-2\log 2}{\log 250}\approx 1.2064$$

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