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Question: How do you use a calculator to evaluate the expression \(\log 0.8\) to four decimal places? \[\]...

How do you use a calculator to evaluate the expression log⁡0.8\log 0.8 to four decimal places? $$$$

Explanation

Solution

We recall the definition of logarithm, argument of logarithm and the common logarithm. In order to find the value of log⁡0.8\log 0.8 we need to type the log⁡\log l button in the scientific calculator and put the value of the argument in the bracket. $$$$

Complete answer:
We know that the logarithm is the inverse operation to exponentiation. That means the logarithm of a given number xx is the exponent to which another fixed number, the base bb must be raised, to produce that numberxx, which means if by=x{{b}^{y}}=x then the logarithm denoted as log and calculated as
log⁡bx=y{{\log }_{b}}x=y
Here xx is called the argument of the logarithm which is always positive. If the base b=10b=10 we call logarithm common logarithm and write without base as log⁡x\log x.
We are asked to evaluate the value of log⁡0.8\log 0.8 using a calculator. We can find logarithm only scientific or graphic calculators. Step−1:Weturnonthecalculator. Step-1:We turn on the calculator.
Step-2:We find the log⁡\log button somewhere in the top or middle. We press it will showlog⁡()\log \left( {} \right) or log⁡(\log \left( {} \right.. Step-3: We type $0.8$ in the bracket as $\log \left( 0.8 \right)$. If there is only one bracket we type $\log \left( 0.8 \right.$ and then close the bracket $\log \left( 0.8 \right)$.
Step-4:We press = or EXE\text{= or EXE} to display the value as -0.0\text{9691}00\text{13}$$$$$ So the value of \log 0.8upto4decimalisup to 4 decimal is-0.0969$

Note: We can also find log⁡0.8\log 0.8 without a calculator with logarithmic identities and the known value log⁡2=0.30102\log 2=0.30102. Let us consider
log⁡0.8=log⁡810\log 0.8=\log \dfrac{8}{10}
We use the identity of quotient log⁡(mm)=log⁡m−log⁡n\log \left( \dfrac{m}{m} \right)=\log m-\log n for m=8,n=10m=8,n=10 in the above step to have;

& \Rightarrow \log 0.8=\log 8-\log 10 \\\ & \Rightarrow \log 0.8=\log {{2}^{3}}-\log 10 \\\ \end{aligned}$$ We use identity of power $\log {{x}^{m}}=m\log x$ for $x=2.m=3$ and the known value $\log 10=1$ in the above step to have ; $$\Rightarrow \log 0.8=3\log 2-1$$ We put the known logarithmic value $\log 2=0.30102$ in the above step to have; $$\Rightarrow \log 0.8=3\times 0.301029-1=0.90306-1=0.0964$$ We can improve the values of $\log 2$ to find with more accuracy.