Aus Online Mathematik Brückenkurs 1

Before we even start thinking about transforming \displaystyle \log _{2} and \displaystyle \log _{3} to ln, we use the log laws

\displaystyle \lg a^{b}=b\centerdot \lg a

\displaystyle \lg \left( a\centerdot b \right)=\lg a+\lg b

to simplify the expression

\displaystyle \begin{align} & \log _{3}\log _{2}3^{118}=\log _{3}\left( 118\centerdot \log _{2}3 \right) \\ & =\log _{3}118+\log _{3}\log _{2}3 \\ \end{align}

With help of the relation

\displaystyle 2^{\log _{2}x}=x and \displaystyle 3^{\log _{3}x}=x

and taking the natural logarithm , we can express \displaystyle \log _{2} and \displaystyle \log _{3} using ln,

\displaystyle \log _{2}x=\frac{\ln x}{\ln 2} and \displaystyle \log _{3}x=\frac{\ln x}{\ln 3}

The two terms log3 118 and log3 log2 3 can therefore be written as

\displaystyle \log _{3}118=\frac{\ln 118}{\ln 3} and \displaystyle \log _{3}\log _{2}3=\log _{3}\left( \frac{\ln 3}{\ln 2} \right),

where we can simplify the last expression further with the logarithm law, log a/b = log a – log b, and then transform \displaystyle \log _{3} to ln,

\displaystyle \begin{align} & \log _{3}\left( \frac{\ln 3}{\ln 2} \right)=\log _{3}\ln 3-\log _{3}\ln 2 \\ & =\frac{\ln \ln 3}{\ln 3}-\frac{\ln 2}{\ln 3} \\ \end{align}

In all, we thus obtain

\displaystyle \log _{3}\log _{2}3^{118}=\frac{\ln 118}{\ln 3}+\frac{\ln \ln 3}{\ln 3}-\frac{\ln 2}{\ln 3}

Input into the calculator gives

\displaystyle \log _{3}\log _{2}3^{118}\approx 4.762

NOTE: the button sequence on a calculator will be:

\displaystyle \begin{align} & \left[ 1 \right]\quad \left[ 1 \right]\quad \left[ 8 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ + \right] \\ & \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ - \right]\quad \left[ 2 \right] \\ & \left[ \text{LN} \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\ & \quad \\ \end{align}