$\log_{10}[\frac1{2^x+x-1}]=x[\log_{10}5-1]$

Question

If $\log_{10}[\frac1{2^x+x-1}]=x[\log_{10}5-1],$ then $x =$

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Answer 1

$$-\log_{10}(2^x+x-1)=x\log_{10}5-x$$ $$\log_{10}(2^x+x-1)=x-\log_{10}5^x$$ $$2^x+x-1=10^{x-\log_{10}5^x}$$ $$2^x+x-1=\frac{10^{x}}{10^{\log_{10}5^x}}$$ $$2^x+x-1=\frac{10^{x}}{5^x}$$ $$2^x+x-1=2^x$$

Therefore, $x=1$.