329/MR00. problem / Math - Solved Math Problems

MR-329. problem

Source: MathReference: Generic Math Textbook - 329.)

Algebra → Exponential equations and inequalities → Exponential equations

x^{\frac{\log_{10} x + 5}{3}} = \frac{10^{5 + \log_{10} x}}{l o g_{10} 10}

x^{\frac{1}{3} (\log_{10} x + 5)} = \frac{10^{5 + \log_{10} x}}{1}

x^{\frac{1}{3} (\log_{10} x + 5)} = 10^{5 + \log_{10} x} l o g_{10} ()

l o g_{10} x^{\frac{1}{3} \cdot (\log_{10} x + 5)} = l o g_{10} 10^{5 + l o g_{10} x}

\frac{1}{3} \cdot (\log_{10} x + 5) \cdot l o g_{10} x = (5 + l o g_{10} x) \cdot l o g_{10} 10

\frac{1}{3} \cdot (\log_{10} x + 5) \cdot \log_{10} x = (5 + \log_{10} x) \cdot 1

\log_{10} x = t

\frac{1}{3} \cdot (t + 5) \cdot t = 5 + t

(t + 5) \cdot t = 15 + 3 t

t_{1, 2} = \frac{- 2 \pm \sqrt{2^{2} - 4 \cdot 1 \cdot (- 15)}}{2}

t_{1, 2} = \frac{- 2 \pm \sqrt{4 + 60}}{2} = \frac{- 2 \pm \sqrt{64}}{2}

t_{1, 2} = \frac{- 2 \pm 8}{2}

t_{1} = 3; t_{2} = - 5

\log_{10} x_{1} = 3

x_{1} = 10^{3} = 1000

\log_{10} x_{2} = - 5

x_{2} = 10^{- 5} = \frac{1}{10^{5}}