Question

How do you use the properties of logarithms to expand ${\log }_5\left({\displaystyle \frac{5}{x}}\right)$?

Answer 1

Hint: Here we will use the logarithmic properties to simplify the given expression. We will use Product rule: ${\log }_a\left({\displaystyle \frac{x}{y}}\right)={\log }_{a}x-{\log }_{a}y$ also apply the tangent and cot identity and then will simplify for the resultant value.

Complete step by step solution:
Take the given expression: ${\log }_5\left({\displaystyle \frac{5}{x}}\right)$
Apply the property: ${\log }_a\left({\displaystyle \frac{x}{y}}\right)={\log }_{a}x-{\log }_{a}y$
$={\log }_{5}5-{\log }_{5}x$
Now, using the base rule: ${\log }_{a}a=1$
$=1-{\log }_{5}x$
Hence, the required solution is ${\log }_5\left({\displaystyle \frac{5}{x}}\right)=1-{\log }_{5}x$

Additional Information:
Also refer to the below properties and rules of the logarithm.
Product rule: ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$
Quotient rule: ${\log }_a{\displaystyle \frac{x}{y}}={\log }_{a}x-{\log }_{a}y$
Power rule: ${\log }_a{x}^n=n{\log }_{a}x$
 Base rule: ${\log }_{a}a=1$
Change of base rule: ${\log }_{a}M={\displaystyle \frac{\log M}{\log N}}$
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
$\ln (x)=\log x÷\log (2.71828)$

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.