Question

How do you evaluate $ {{\log }_{0.8}}\left( 12 \right)$?

Answer 1

Hint: For this question, we can solve by using the properties of log that is $ {{\log }_{a}}b=\dfrac{\ln b}{\ln a}$. Here, a = 0.8 and b = 12 . Then, we should calculate the log values separately. And finally divide both of them to get the final solution for the given equation.

Complete step by step solution:
According to the problem, we are asked to evaluate the equation $ {{\log }_{0.8}}\left( 12 \right)$--- ( 1 )
Therefore, by using the property of log that is $ {{\log }_{a}}b=\dfrac{\ln b}{\ln a}$, in equation 1, we get:
Here , a = 0.8 and b = 12 .
Therefore, we can get:
$\Rightarrow {{\log }_{0.8}}12=\dfrac{\ln 12}{\ln 0.8}$ ---- (2)
Now, we calculate the value of ln(12).
When we calculate the value of ln(12), we get:
$\Rightarrow \ln 12=2.48490665$ --(3)

Now, we calculate the value of ln(0.8).
When we calculate the value of ln(0.8), we get:
$\Rightarrow \ln \left( 0.8 \right)=-0.223143551$ --(4)

After substituting equation 3 and equation 4 in equation 2, we get
$\Rightarrow {{\log }_{0.8}}12=\dfrac{\ln 12}{\ln 0.8}$

$\Rightarrow {{\log }_{0.8}}12=\dfrac{2.48490665}{-0.223143551}$

$\Rightarrow {{\log }_{0.8}}12=-11.1359107$ ------ Final answer

So, by evaluating the given question that is $ {{\log }_{0.8}}\left( 12 \right)$ we get the answer as -11.1359107 .
Therefore, the solution of the given equation ${{\log }_{0.8}}\left( 12 \right)$ is ${{\log }_{0.8}}12=-11.1359107$.

Note: It is very important to remember the logarithmic properties So that we could directly apply the properties or the formulas in the equation and directly get the answers. For this question, we could also verify the derived answer by using 0.8 to the power of the answer. If we get the question as the answer, then the answer we got is correct. We should do this as ${{0.8}^{-11.1359107}}=12$.