## College Algebra (11th Edition)

$M=M_0\left(10^{\frac{6-m}{2.5}}\right)$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $m=6-2.5\log \left( \dfrac{M}{M_0} \right) ,$ in terms of $M ,$ use the properties of equality to isolate the logarithmic expression. Then change to exponential form. Finally use again the properties of equality to isolate the needed variable. $\bf{\text{Solution Details:}}$ Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} 2.5\log \left( \dfrac{M}{M_0} \right)=6-m \\\\ \dfrac{2.5\log \left( \dfrac{M}{M_0} \right)}{2.5}=\dfrac{6-m}{2.5} \\\\ \log \left( \dfrac{M}{M_0} \right)=\dfrac{6-m}{2.5} .\end{array} Since $\log_by=x$ is equivalent to $y=b^x$, the equation above, in exponential form, is equivalent to \begin{array}{l}\require{cancel} \log_{10} \left( \dfrac{M}{M_0} \right)=\dfrac{6-m}{2.5} \\\\ \dfrac{M}{M_0}=10^{\frac{6-m}{2.5}} .\end{array} Using the properties of equality to isolate the needed variable results to \begin{array}{l}\require{cancel} M_0\cdot\dfrac{M}{M_0}=M_0\cdot10^{\frac{6-m}{2.5}} \\\\ M=M_0\left(10^{\frac{6-m}{2.5}}\right) .\end{array}