Question

Use change of base logarithmic rule for \[\log _{0.008}5\]

Answer 1

Hint: First we will see the useful things about the logarithmic functions, which is the exponent of the given function.
The logarithmic form is written as ${\log _3}9 = 2$ where two is the exponent and the base value is three which is given below in that log function.
We are also able to say that the logarithmic of the nine to the base value of three is two.
Formula used: General logarithm function will be defined as the $f(x) = {\log _b}x$ where b is the base.

Complete step by step answer:
First, let us take the given log function\[\log _{0.008}5\], now applying the given value into the logarithm formula we get \[\log _{0.008}5 = x\](equals to anything of the values or variable or even polynomial as in the hint).
As we said, the logarithm base is an exponent function, so converting the equation we get\[{5^x} = 0.008\].
Now turning the decimal terms into the ration numbers, we get \[{5^x} = \dfrac{8}{{1000}}\] and with the help of the division, the method divides the common terms, \[{5^x} = \dfrac{1}{{125}}\] (the common term is eight).
Rewriting the equation into the cubic form of the three, we get \[{5^x} = \dfrac{1}{{{5^3}}}\] (to get the number three on the right-hand side we did that).
Again, by the use of division and inverse image function, we get \[{5^x} = {5^{ - 3}}\] (inverse power will be changed to opposite to the given).
Thus, both values of five will be canceled and thus we get\[x = - 3\].
Hence the value of the logarithmic \[\log _{0.008}5\] using the base rule we get\[x = - 3\].

Note: the product of the log rule is $\log (ab) = \log a + \log b$.
The quotient rule of the log is $\log (\dfrac{a}{b}) = \log a - \log b$.
Logarithm power rule is $\log ({x^a}) = a\log x$, which is also the step we applied in above simplifying.
The zero exponents of the log are ${\log _a}1 = 0$.