From Förberedande kurs i matematik 1

The calculator does not have button for \displaystyle \log _{3}, but it does have one for the natural logarithm ln, so we need to rewrite \displaystyle \log _{3}4 in terms of ln.

If we go back to the definition of the logarithm, we see that \displaystyle \log _{3}4 is that number which satisfies

\displaystyle 3^{\log _{3}4}=4

Now, take the natural logarithm of both sides,

\displaystyle \ln 3^{\log _{3}4}=\ln 4

Using the logarithm law, \displaystyle \lg a^{b}=b\lg a, the left-hand side can be written as \displaystyle \log _{3}4\centerdot \ln 3 and the relation is

\displaystyle \log _{3}4\centerdot \ln 3=\ln 4

Thus, after dividing by \displaystyle \text{ln 3}, we have

\displaystyle \log _{3}4=\frac{\ln 4}{\ln 3}=\frac{1.386294...}{1.098612...}=1.2618595

which gives 1.262 as the rounded-off answer.

NOTE: on a calculator, the answer is obtained by pressing the buttons

\displaystyle \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right]