3
$\begingroup$

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 ?

$\endgroup$
  • 1
    $\begingroup$ I find it very doubtful that you're not allowed to involve the use of a calculator at any point.. Are you sure you're not being asked to evaluate it in a way that is feasible with a calculator? The average calculator can't handle $3^{750}/5^{510}$, but it can certainly handle $10^{(750\log3 - 510\log5)}$ $\endgroup$ – AlkaKadri Jan 15 '18 at 22:17
2
$\begingroup$

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 $23.24$

$\endgroup$
1
$\begingroup$

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 $750\log_5 3 \approx 511.95.$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.