Answer
no solution
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\log(x-10)-\log(x-6)=\log2
,$ use the properties of logarithms to simplify the left-hand expression. Then drop the logarithm on both sides. Use the properties of equality to isolate the variable. Finally, do checking of the solution with the original equation.
$\bf{\text{Solution Details:}}$
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent \begin{array}{l}\require{cancel} \log\dfrac{x-10}{x-6}=\log2 .\end{array}
Since the logarithm on both sides have the same base, then the logarithm can be dropped. Hence, the equation above is equivalent to \begin{array}{l}\require{cancel} \dfrac{x-10}{x-6}=2 .\end{array}
Since $\dfrac{a}{b}=\dfrac{c}{d}$ implies $ad=bc$ or sometimes referred to as cross-multiplication, the equation above is equivalent to \begin{array}{l}\require{cancel} \dfrac{x-10}{x-6}=\dfrac{2}{1} \\\\ (x-10)(1)=(x-6)(2) \\\\ x-10=2x-12 .\end{array}
Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} x-10=2x-12 \\\\ x-2x=-12+10 \\\\ -x=-2 \\\\ x=2 .\end{array}
If $
x=2
,$ the part of the given expression, $
\log(x-10)
,$ becomes $
\log(-8)
.$ This is not allowed since $
\log x$ is defined only for positive values of $x.$ Hence, there is $\text{
no solution
.}$
