Exponents and Logarithms – Practice questions

Q1. If  $3^x=5^{x–1}$  , find the value of  $a$  if  $x$  can be written as ${\log }_a\left(\frac{1}{5}\right)$ .

Q2. Solve for  $x$  : $e^{2\ln \left(\ln x\right)}=2–3\ln x–{\left(\ln x\right)}^2$ .

Q3. Solve for  $x$  :  ${\log }_{\left(x–2\right)}\left(x–1\right)+1>{\log }_{\left(x–2\right)}6$ .

Q4. Solve for  $x$  :  ${\log }_{2}x {\log }_{3}x {\log }_{4}x={\log }_{2}x {\log }_{3}x+{\log }_{3}x {\log }_{4}x+{\log }_{4}x {\log }_{2}x$ .

Q5. If  ${\log }_{70}5=a$  ,  ${\log }_{70}7=b$ find the value of    ${\log }_{70}\left(\frac{1}{8}\right)$ in terms of  $a$   and  $b$  .

Q6. Solve for  $x$  :  ${\log }_{9}x={\log }_3\left(x–2\right)$  .

Q7. Find the value of  $x$  for which  $x^{{\log }_{3}x}>3$  .

Q8. Find the solution of the equation  $3^x=x^3$  .

Q9. Solve for  $x$  : $\frac{1}{2}\log \left(x+5\right)=\log \left(x–1\right)$ .

Q10. Find the value of  $x$ if: ${\log }_3\left(10–3^x\right)=2–x$ .

Q11. If  ${\log }_9\left( {\log }_{3}x\right)+{\log }_3\left({\log }_{27}x\right)=2$ , then find the value of  $x$  in the form of  $a^b$ , where  $a, b\in {\mathrm{\mathbb{Z}}}^{+}$ .