Question 383402
I'm guessing you posted this yesterday and I "solved" it for you. But I misread the problem and solved something else. Sorry about that!<br>
If your problem is to find the number for {{{3*log(3, (7))}}} (or {{{log(3, (7^3))}}} or {{{log(3, (343))}}}), without a calculator, then the answer is: <i>It can't be done!</i> (This leads me to suspect that problem may actually be the problem I thought it was yesterday.)<br>
7 is not a well-known power of 3, 3 is not a well-known power of 7 and both numbers, 3 and 7, are not well-known powers of some other number. One of these would have to be true in order to answer this question without a calculator.<br>
If the 7 was a 9 instead then we could solve this problem because 9 is a well-known power of 3. Since {{{9 = 3^2}}}, {{{log(3, (9)) = 2}}} and {{{3*log(3, (9)) = 3*2 = 6}}}<br>
The only thing you can do with your expression is to convert it to an expression of logarithms of a different base. For example, if you did want to use your calculator to find a decimal approximation for the value of your expression, then we could use the base conversion formula for logarithms, {{{log(a, (p)) = log(b, (p))/log(b, (a))}}} to convert your base 3 logarithm into a base that your calculator "knows" like base 10 or base e (aka ln). Converting your base 3 logarithm into base e logarithms we get:
{{{log(3, (7)) = ln(7)/ln(3)}}}
which would make your full expression:
{{{3*log(3, (7)) = 3*(ln(7)/ln(3))}}}
With this we could then use our calculator to obtain a decimal approximation for the value of your expression.