Question

If 10^0.3010 = 2, then find the value of log0.125 (125) ?

Answer 1

It is given that $10^{0.3010}=2$

Therefore, log to the base 10 on both sides will give us:

$0.3010log_{10}10=log_{10}2$

or $0.3010=log_{10}2$...................Equation 1

Now, we have also been given the expression $log_{0.125}125$. Let us equate this expression to the real number n. Thus, we will have:

$log_{0.125}125=n$

By the definition of logarithms, this would mean that:

0.125^n=125.............Equation 2

Now, $0.125=\frac{1}{2^3}$

Therefore, Equation 2 can be rewritten as:

$(\frac{1}{2^3})^n=125$

Therefore, 2^{-3n}=125

Taking log of the above equation to the base 10 we will get:

$-3nlog_{10}2=log_{10}125$..........Equation 3

Using Equation 1 and Equation 3 we get:

$-3n\times 0.3010=log_{10}125\approx2.097$

Therefore, $n=\frac{2.097}{-3\times 0.3010} \approx -2.322$

Therefore, $log_{0.125}125=n=-2.322$