Evaluating the quotient without the calculator Without using the calculator find the value of 
$$\frac{3^{750}}{5^{510}}$$
I tried to consider 
$$\frac{3^{750}}{5^{510}}=x $$
Then i took log to both sides to be
$$750\log{3}-510\log{5}=\log{x}$$
But i could not do any thing more ?
 A: This is a good start.  If you are permitted log tables, you can look up $\log 3$ and $\log 5$ and do the calculation you indicate, then use the tables to get $x$.  Otherwise, I don't see any way.  It turns out that $\frac {3^{747}}{5^{510}}$ is just slightly less than $1$ and the final answer is about $23.24$
A: Supposing you're also not allowed logarithm tables, let's see if we can get some crude guesses about logarithms:
$$
\begin{array}{ccc}
n & 3^n & 5^n \\
\hline
1 & 3 & 5 \\
2 & 9 & 25 \\
3 & 27 & 125 \\
\vdots & \vdots & \vdots
\end{array}
$$
So we have $3^3 \approx 5^2,$ so $5^{2/3} \approx 3,$ so $\log_5 3 \approx 2/3.$
But $3^3>5^2,$ so $5^{2/3} < 3,$ so $\log_5 3 >2/3. $
$$
(750\log_5 3) - 510 \approx \frac 2 3 \cdot750 - 510 = 500 - 510 \approx \cdots
$$
But $2/3$ was a bit too small, so maybe around $510 - 510\text{ ?}$ However, that cannot be exact, because $3$ and $5$ are prime numbers, so the numerator and denominator cannot have any factors that cancel each other. If the logarithm were $0$ then the quotient would be $1.$ So it's probably not far from $1.$
With a calculator I'm getting $750\log_5 3 \approx 511.95.$
