Question

The value of ${\log _{0.2}}5$ is equal to:
A) $( - 1)$
B) $1$
C) $2$
D) $ - 2$

Answer 1

Hint: Logarithms are ways to figure out which exponents we need to multiply into the specific number. First of all we will assume that the given expression is equal to “x”. Here, using the property of the inverse logarithm, is use the formula as $y = {\log _b}x \Rightarrow {b^y} = x$.

Complete step by step solution:
Take the given expression: ${\log _{0.2}}5$
Let us assume that the above expression is equal to ‘x”
$ \Rightarrow {\log _{0.2}}5 = x$
By using the inverse properties of logarithm –
$\Rightarrow 5 = {(0.2)^x}$
Simplify the above expression –
$\Rightarrow 5 = {\left( {\dfrac{2}{{10}}} \right)^x}$
Common factors from the numerator and the denominator cancels each other.
$\Rightarrow 5 = {\left( {\dfrac{1}{5}} \right)^x}$
By using the inverse property, or reciprocal property – the above expression can be re-written as –
$\Rightarrow 5 = {\left( {{5^{ - 1}}} \right)^x}$
Simplify the above expression –
$\Rightarrow 5 = {\left( 5 \right)^{ - x}}$
When bases are the same, powers are equal.
$ \Rightarrow - x = 1$
Multiplying with $( - 1)$ on both the sides of the above equation –
$ \Rightarrow x = - 1$

Thus the correct answer is option ‘A’.

Additional Information: Know the difference between ln and log and apply its properties accordingly. Logarithms are the ways to figure out which exponents we need to multiply into the specific number. Log is defined for the base $10$ and ln is denoted for the base e. “e” is an irrational and transcendental number which can be expressed as $e = 2.71828$. You can convert ln to log by using the relation such as $\ln (x) = \log x \div \log (2.71828)$

Note: In other words, the logarithm is the power to which the number must be raised in order to get some other. Always remember the standard properties of the logarithm. Product rule, quotient rule and the power rule. The basic logarithm properties are most important and the solution solely depends on it, so remember and understand its application properly. Be good in multiples and know the concepts of square and square root and apply accordingly.