## College Algebra (11th Edition)

$x=3-\sqrt[4]{1000}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $\log(3-x)=0.75 ,$ convert to exponential form. Then use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Since $\log x=\log_{10} x,$ the equation above is equivalent to \begin{array}{l}\require{cancel} \log_{10}(3-x)=0.75 .\end{array} Since $y=b^x$ is equivalent to $\log_b y=x,$ the exponential form of the equation above is \begin{array}{l}\require{cancel} 10^{0.75}=3-x .\end{array} Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} x=3-10^{0.75} \\\\ x=3-10^{\frac{3}{4}} .\end{array} Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} x=3-\sqrt[4]{10^3} \\\\ x=3-\sqrt[4]{1000} .\end{array}