Question

How do you solve $\log \left( x \right) = - 0.123$?

Answer 1

In order to solve this we need to know the property of the logarithm and there are many such properties but we need to use them according to our need. We will use the property which says that:
If ${\log _a}b = c$ then we can say that ${a^c} = b$.

Complete step by step solution:
Here we are given to solve the logarithmic function which is given as $\log \left( x \right) = - 0.123$.
If we are given ${\log _a}b = c$ then we must know that here $a$ is the base of the logarithm.
Here we must know that if we are given $\ln x$ then its base is taken as $e = 2.718$ and if it is simply given as $\log x$ then the base is actually taken as $10$ over here. Hence we must be clear with this point.
Now we need to solve for $\log \left( x \right) = - 0.123$ which means we need to find the value of $x$.
We know the property of $\log$ which says that:
If ${\log _a}b = c$ then we can say that ${a^c} = b$.
Now we know that in the above problem we have to take the base at $10$.
So we can compare ${\log _a}b = c$ with the given logarithmic function which is $\log \left( x \right) = - 0.123$.
So we will get:
$a = 10 \\\ b = x \\\ c = - 0.123 \\\$
So we know that if ${\log _a}b = c$ then we can say that ${a^c} = b$
So we can write ${10^{ - 0.123}} = x$
$x = {10^{ - 0.123}}$
Now we can solve this with the calculator and get the value as:
$x = {10^{ - 0.123}} = 0.7534$.

Note: Whenever the student is given to solve the problems which contain logarithmic function, he must know the properties of the log and also when to use which formula. Hence this is very necessary and it comes by practice. The properties of $\log$ are like:
$\log (ab) = \log a + \log b \\\ \log \left( {\dfrac{a}{b}} \right) = \log a - \log b \\\$
$\log {m^n} = n\log m$.