# Solving logarithmic equations without calculator

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 $x = 1/10$.

• Do you mean $\log_x 10=5 (\log x)+4$? – zz20s Jan 15 '16 at 3:48
• I edited the question – Melanie Jan 15 '16 at 3:48
• Can you use mathjax? It would make your equation far easier to read. – zz20s Jan 15 '16 at 3:48
• Yeah I will and it's the equation you suggested. – 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 $\log_{a} a=1$.