To evaluate `log_3 314` , use the formula for change of base of logarithms:

`log_b a = (log_c a)/(log_c b)`

Then the given logarithm can be evaluated as

`log_3 314 = (ln 314)/(ln_3)` (ln is the natural logarithm, or logarithm with the base e). Natural logarithms can be found using a calculator.

`log_3 314 = 5.74939/1.09861 =5.2333 `

**The value of the given logarithm rounded to 4 decimal places is 5.2333.**

Step 1: Rewrite as 3^x = 314

Step 2: Then, use Laws of Logs to write as follows:

x log 3 = log 314

Step 3: Use change of base rule

Divide both sides of the equation by log 3.

x (log 3/log3) = log 314/log 3

**The answer is 5.2333.**

Log(base 3) 314 = 5.2333

(Found using online logarithm calculator)

http://www.1728.org/logrithm.htm

We could guess that the answer would be approximately more than 5, because 3^5 = 243 and 3^6 = 729.

We need to find log(base 3) 314 and round to 4 decimal places.

A standard graphing calculator can do this automatically but lets say you don't have one. We can just use the change of base formula which is super easy:

Your problem becomes : log314 / log3

which is 5.2333