Solution 3.3:6c
From Förberedande kurs i matematik 1
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- | {{ | + | Before we even start thinking about transforming <math>\log_2</math> and <math>\log_3</math> to ln, we use the log laws |
- | < | + | |
- | {{ | + | {{Displayed math||<math>\begin{align} |
- | {{ | + | \log a^b &= b\cdot\log a\,,\\[5pt] |
- | < | + | \log (a\cdot b) &= \log a+\log b\,, |
- | {{ | + | \end{align}</math>}} |
+ | |||
+ | to simplify the expression | ||
+ | |||
+ | {{Displayed math||<math>\begin{align} | ||
+ | \log_{3}\log _{2}3^{118} | ||
+ | &= \log_{3}(118\cdot\log_{2}3)\\[5pt] | ||
+ | &= \log_{3}118 + \log_{3}\log_{2}3\,\textrm{.} | ||
+ | \end{align}</math>}} | ||
+ | |||
+ | With help of the relation <math>2^{\log_{2}x} = x</math> and <math>3^{\log_{3}x} = x</math> and taking the natural logarithm , we can express <math>\log_{2}</math> and <math>\log_{3}</math> using ln, | ||
+ | |||
+ | {{Displayed math||<math>\log_{2}x=\frac{\ln x}{\ln 2}\quad</math> and <math>\quad\log_{3}x = \frac{\ln x}{\ln 3}\,\textrm{.}</math>}} | ||
+ | |||
+ | The two terms <math>\log_3 118</math> and <math>\log_3\log_2 3</math> can therefore be written as | ||
+ | |||
+ | {{Displayed math||<math>\log_{3}118 = \frac{\ln 118}{\ln 3}\quad</math> and <math>\quad\log_{3}\log_{2}3 = \log_{3}\frac{\ln 3}{\ln 2}\,,</math>}} | ||
+ | |||
+ | where we can simplify the last expression further with the logarithm law, log (a/b) = log a – log b, and then transform <math>\log _{3}</math> to ln, | ||
+ | |||
+ | {{Displayed math||<math>\begin{align} | ||
+ | \log_{3}\frac{\ln 3}{\ln 2} | ||
+ | &= \log_{3}\ln 3 - \log_{3}\ln 2\\[5pt] | ||
+ | &= \frac{\ln\ln 3}{\ln 3} - \frac{\ln\ln 2}{\ln 3}\,\textrm{.} | ||
+ | \end{align}</math>}} | ||
+ | |||
+ | In all, we thus obtain | ||
+ | |||
+ | {{Displayed math||<math>\log_{3}\log_{2}3^{118} = \frac{\ln 118}{\ln 3} + \frac{\ln \ln 3}{\ln 3} - \frac{\ln\ln 2}{\ln 3}\,\textrm{.}</math>}} | ||
+ | |||
+ | Input into the calculator gives | ||
+ | |||
+ | {{Displayed math||<math>\log_{3}\log_{2}3^{118}\approx 4\textrm{.}762\,\textrm{.}</math>}} | ||
+ | |||
+ | |||
+ | Note: The button sequence on the calculator will be: | ||
+ | |||
+ | |||
+ | <center> | ||
+ | {| | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|1 | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|1 | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|8 | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|÷ | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|3 | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|+ | ||
+ | |} | ||
+ | |- | ||
+ | |height="7px"| | ||
+ | |- | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|3 | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|÷ | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|3 | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
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+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|- | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|2 | ||
+ | |} | ||
+ | |- | ||
+ | |height="7px"| | ||
+ | |- | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|÷ | ||
+ | |} | ||
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+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|3 | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|LN | ||
+ | |} | ||
+ | || | ||
+ | || | ||
+ | {| border="1" cellpadding="3" cellspacing="0" | ||
+ | |width="30px" align="center"|= | ||
+ | |} | ||
+ | |} | ||
+ | </center> |
Current revision
Before we even start thinking about transforming \displaystyle \log_2 and \displaystyle \log_3 to ln, we use the log laws
\displaystyle \begin{align}
\log a^b &= b\cdot\log a\,,\\[5pt] \log (a\cdot b) &= \log a+\log b\,, \end{align} |
to simplify the expression
\displaystyle \begin{align}
\log_{3}\log _{2}3^{118} &= \log_{3}(118\cdot\log_{2}3)\\[5pt] &= \log_{3}118 + \log_{3}\log_{2}3\,\textrm{.} \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}\quad and \displaystyle \quad\log_{3}x = \frac{\ln x}{\ln 3}\,\textrm{.} |
The two terms \displaystyle \log_3 118 and \displaystyle \log_3\log_2 3 can therefore be written as
\displaystyle \log_{3}118 = \frac{\ln 118}{\ln 3}\quad and \displaystyle \quad\log_{3}\log_{2}3 = \log_{3}\frac{\ln 3}{\ln 2}\,, |
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}\frac{\ln 3}{\ln 2} &= \log_{3}\ln 3 - \log_{3}\ln 2\\[5pt] &= \frac{\ln\ln 3}{\ln 3} - \frac{\ln\ln 2}{\ln 3}\,\textrm{.} \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\ln 2}{\ln 3}\,\textrm{.} |
Input into the calculator gives
\displaystyle \log_{3}\log_{2}3^{118}\approx 4\textrm{.}762\,\textrm{.} |
Note: The button sequence on the calculator will be:
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