Problem: If 10m=b−10n\log _{10} m=b-\log _{10} nlog10m=b−log10n, then m=m=m=
Answer Choices:
A. bn\dfrac{b}{n}nb B. bnb nbn C. 10bn10^{b} n10bn D. b−10nb-10^{n}b−10n E. 10bn\dfrac{10^{b}}{n}n10b
Solution:
b=10(m)+10(n)b=10(mn)10b=mn∴m=10bn\begin{aligned} b &= \log_{10}(m) + \log_{10}(n) \\ b &= \log_{10}(mn) \\ 10^b &= mn \\ \therefore m &= \boxed{\frac{10^b}{n}} \end{aligned} bb10b∴m=log10(m)+log10(n)=log10(mn)=mn=n10b
or\textbf{or} or
10(m)=10(10b)−10(n)=10(10bn)∴m=10bn\begin{aligned} \log_{10}(m) &= \log_{10}(10^b) - \log_{10}(n) \\ &= \log_{10}\left(\frac{10^b}{n}\right) \\ \therefore m &= \boxed{\frac{10^b}{n}} \end{aligned} log10(m)∴m=log10(10b)−log10(n)=log10(n10b)=n10b