Question

Find the value of ${\log _5}0.008$.

Answer 1

Here we need to find the value of the given logarithmic expression. We will first assume the given expression to be any variable and then we will use the different logarithmic properties to simplify the given logarithmic expression. Then we will use the exponential properties to further simplify the expression. After simplifying the terms, we will get the value of the variable and hence the value of the given logarithmic expression.

Formula used:
When $x = {\log _b}a$, then ${b^x} = a$

Complete step by step solution:
Here we need to find the value of the given logarithmic expression and the given expression is${\log _5}0.008$.
Let $x = {\log _5}0.008$
We know from the property of the logarithmic function that:
When $x = {\log _b}a$, then ${b^x} = a$
Using the same property, we get
$\Rightarrow {5^x} = 0.008$
Now, we will convert the decimal into the fractional form.
$\Rightarrow {5^x} = \dfrac{8}{{1000}}$
Now, we will reduce the given fraction further.
$\Rightarrow {5^x} = \dfrac{1}{{125}}$
Now, we will write the denominator as the product of its factors.
$\Rightarrow {5^x} = \dfrac{1}{{5 \times 5 \times 5}} = \dfrac{1}{{{5^3}}}$
We know the property of the exponential function that
$\dfrac{1}{{{a^x}}} = {a^{ - x}}$
Using the same property of logarithmic function, we get
$\Rightarrow {5^x} = {5^{ - 3}}$
We know the properties of the exponential function that
When ${b^x} = {b^y}$, then $x = y$
Using the same property of logarithmic function, we get
$\Rightarrow x = - 3$

Therefore, the value of the given logarithmic expression is equal to -3.
$\Rightarrow {\log _5}0.008 = - 3$

Note: Here we have obtained the value of the given logarithmic expression and we have used various properties of the logarithmic function. Here the logarithmic function is defined as the function which is the inverse of the exponential function i.e. if we take the inverse of the logarithmic function, then we will get the logarithmic function.