Revise what logarithms are and how to use the 'log' buttons on a scientific calculator
Part ofMathsAlgebraic and trigonometric skills
A scientific calculator has two 'log' buttons on it. These are marked log and ln.
The log key is used for calculations of the form \({\log _{10}}x\).
For example, to work out that \(\log 10000 = 4\) you would press \(\log 10000 =\)
Using this method you should be able to find that \(\log 45 = 1.653\)
The ln key is used for calculations of the form \({\log _e}x\).
This is called the natural logarithm. In mathematics the natural logarithm \({\log _e}x\) is usually written as \(\ln x\).
Like \(\pi\), \(e\) is a mathematical constant and has many applications in mathematics, particularly with logs.
For example, to find \({\log _e}2 = 0.693\) , press \(\ln 2\) on your calculator.
Using this method you should find that \({\log _e}\frac{1}{3} = - 1.0986\)
Solve:
In each case rewrite the statement using:
\({a^y} = x \leftrightarrow {\log _a}x = y\)
Solve \({5^x} = 4\)
Take logs of both sides (either base \(e\) or base \(10\))
So \({\log _{10}}({5^x}) = {\log _{10}}4\)
\(\Rightarrow x{\log _{10}}5 = {\log _{10}}4\)
\(\Rightarrow \frac{{{{\log }_{10}}4}}{{{{\log }_{10}}5}}\) which is approximately 0.861
Evaluate \({\log _3}2\)
Let \(x = {\log _3}2\)
Then \({3^x} = 2\)
Now take the log of both sides (base \(e\) or \(10\))
So \({\log _e}{3^x} = {\log _e}2 \Rightarrow x\ln 3 = \ln 2\)
\(\Rightarrow x = \frac{{\ln 2}}{{\ln 3}}\) which is approximately \(0.631\)
(Check \(\frac{{{{\log }_{10}}2}}{{{{\log }_{10}}3}}\) gives the same)
