Hi I am stuck on this question $$ \log_x 10= 5 (\log_{10} x) +4 $$ The answer key gives the solutions $x = 10^{1/5}$ and $x = 1/10$.

  • $\begingroup$ Do you mean $\log_x 10=5 (\log x)+4$? $\endgroup$ – zz20s Jan 15 '16 at 3:48
  • $\begingroup$ I edited the question $\endgroup$ – Melanie Jan 15 '16 at 3:48
  • $\begingroup$ Can you use mathjax? It would make your equation far easier to read. $\endgroup$ – zz20s Jan 15 '16 at 3:48
  • $\begingroup$ Yeah I will and it's the equation you suggested. $\endgroup$ – Melanie Jan 15 '16 at 3:50

If you meant $\log_x 10=5 \log x+4$ and the RHS logarithm is base $10$,

Use the change of base formula to write $\frac{\log 10}{\log x}=5 \log x +4.$

Then write $0= 5\log^2 x +4 \log x -1$. Set $t=\log x$, giving a quadratic $0=5t^2+4t-1$. Can you solve the quadratic?

Edit: Note that $\log_{a} a=1$.


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