## Precalculus (6th Edition) Blitzer

The value of ${{\log }_{2}}x+{{\log }_{2}}\left( 2x-3 \right)=1$ is $x=2$.
We have to find the value of x from ${{\log }_{2}}x+{{\log }_{2}}\left( 2x-3 \right)=1$ by evaluating the left-hand side as follows: \begin{align} & {{\log }_{2}}x+{{\log }_{2}}\left( 2x-3 \right)=1 \\ & {{\log }_{2}}x\left( 2x-3 \right)=1 \\ & {{\log }_{2}}\left[ 2{{x}^{2}}-3x \right]=1 \end{align} Then, using if ${{p}^{q}}=r$, then ${{\log }_{p}}r=q$, one gets: \begin{align} & 2{{x}^{2}}-3x={{2}^{1}} \\ & 2{{x}^{2}}-3x-2=0 \\ & 2{{x}^{2}}-4x+x-2=0 \end{align} Simplify it further: \begin{align} & 2x\left( x-2 \right)+x-2=0 \\ & \left( 2x+1 \right)\left( x-2 \right)=0 \end{align} So, $x=2$ or $x=-\frac{1}{2}$. If $x=-\frac{1}{2}$, then ${{\log }_{2}}x$ does not exist as the logarithm function does not take on negative values. Therefore, the only possible value of x is 2. Thus, the value is $x=2$.