Question

Solve (log(7x)) log x = 5

Logarithms
Solve $$\displaystyle{\left({\log{{\left({7}{x}\right)}}}\right)}{\log{{x}}}={5}$$

2021-08-21

We have to solve the following equation
$$\displaystyle{\left({\log{{\left({7}{x}\right)}}}\right)}{\log{{\left({x}\right)}}}={5}$$
By applying log rule $$\log c(ab)=\log c(a)+\log c(b)$$ we get
$$(\log(7x))\log(x)=5$$

⟹  $$(\log(x)+\log(7))\log(x)=5$$

⟹  $$(u+\log(7))u=5$$,[where $$u=\log(x)$$]

⟹ $$u_2+\log(7)u−5=0$$
The above is a quadratic equation of the form $$au^2+bu+c=0$$. By using quadratic formula we get
$$\displaystyle{u}=\frac{{-{b}\pm\sqrt{{{b}^{{2}}-{4}{a}{c}}}}}{{2}}{a}=\frac{{-{\log{{7}}}\pm\sqrt{{{\log{{7}}}}}^{{2}}+{20}}}{{2}}$$
Since $$\displaystyle{u}={\log{{x}}}$$, x=eu. Therefore, the required solutions are
$$\displaystyle{x}={e}^{{-{\log{{7}}}\pm\frac{\sqrt{{{\left({\log{{7}}}^{{2}}\right)}+{20}}}}{{2}}}}$$