Question

Evaluate log base 3(400) to five decimal places

Answer 1

Evaluate log base 3(400) to five decimal places

First, this problem requires a calculator or computer to solve. The trouble is that most (all?) calculators do not do log base 3. They do know log base 10 (labeled log) and log base e (labeled ln). We will use ln (natural log)

To solve this problem, it is very important to know that
\[ log_3(400)=x \]
is the same problem as
\[ 400 = 3^{x} \]
If this seems strange, you might want to watch
http://www.khanacademy.org/math/algebra/logarithms/v/introduction-to-logarithms

Take the ln of both sides
\[ ln(400)= ln(3^{x}) \]

Now use the exponent rule of logs
(See
http://www.khanacademy.org/math/algebra/logarithms/v/logarithm-of-a-power
)
\[ ln(400) = xln(3) \]
Divide both sides by ln(3):
\[ \frac{ln(400)}{ln(3)}=x \]

If you cut and paste
ln(400)/ln(3)=
into the google window it will calculate the answer:
5.45366606

Finally, round the answer to 5 decimal places. Count to the 6th decimal
place
123456
5.453666

It is 6, (5 or bigger) so make the 5th decimal one bigger:
5.45367

That is the answer.