Combine form

Algebra Level 3

$\large x^{\log_{10} 5} + 5^{\log_{10} x} = 50$

Solve for real $x$ satisfying the equation above.

The answer is 100.

1 solution

Chew-Seong Cheong
Jan 31, 2017

$\begin{aligned} x^{\log_{10} 5} + {\color{#3D99F6}5^{\log_{10} x}} & = 50 & \small \color{#3D99F6} a^{\log b} = b^{\log a} \\ x^{\log_{10} 5} + {\color{#3D99F6}x^{\log_{10} 5}} & = 50 \\ 2 x^{\log_{10} 5} & = 50 \\ x^{\log_{10} 5} & = 25 \\ \log_{10} \left(x^{\log_{10} 5}\right) & = \log 25 \\ \log_{10} 5 \log_{10} x & = 2 \log_{10} 5 \\ \log_{10} x & = 2 \\ \implies x & = \boxed{100} \end{aligned}$

Combine form

The answer is 100.

1 solution

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