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5-lnx-50-x-ln-5-




Question Number 81437 by jagoll last updated on 13/Feb/20
5^(lnx)  = 50 −x^(ln 5)
$$\mathrm{5}\:^{\mathrm{lnx}} \:=\:\mathrm{50}\:−\mathrm{x}^{\mathrm{ln}\:\mathrm{5}} \\ $$$$ \\ $$
Commented by john santu last updated on 13/Feb/20
formula a^(log_b  c)  = c^(log_b  a)   ⇒ 5^(ln x )  = 50 − 5^(ln x)   2.5^(ln x)  = 50 ⇒ 5^(ln x)  = 5^(ln x)  = 5^2   ln x = 2 ⇒ e^(ln x)  = e^2    ∴ x = e^2  .
$$\mathrm{formula}\:\mathrm{a}^{\mathrm{log}_{\mathrm{b}} \:\mathrm{c}} \:=\:\mathrm{c}^{\mathrm{log}_{\mathrm{b}} \:\mathrm{a}} \\ $$$$\Rightarrow\:\mathrm{5}^{\mathrm{ln}\:\mathrm{x}\:} \:=\:\mathrm{50}\:−\:\mathrm{5}^{\mathrm{ln}\:\mathrm{x}} \\ $$$$\mathrm{2}.\mathrm{5}^{\mathrm{ln}\:\mathrm{x}} \:=\:\mathrm{50}\:\Rightarrow\:\mathrm{5}^{\mathrm{ln}\:\mathrm{x}} \:=\:\mathrm{5}^{\mathrm{ln}\:\mathrm{x}} \:=\:\mathrm{5}^{\mathrm{2}} \\ $$$$\mathrm{ln}\:\mathrm{x}\:=\:\mathrm{2}\:\Rightarrow\:\mathrm{e}^{\mathrm{ln}\:\mathrm{x}} \:=\:\mathrm{e}^{\mathrm{2}} \: \\ $$$$\therefore\:\mathrm{x}\:=\:\mathrm{e}^{\mathrm{2}} \:. \\ $$

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