Question

How do you evaluate ${\log _5}625$?

Answer 1

Hint: A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.
 $y = {b^x}$exponential form,
$x = {\log _b}y$ logarithmic function, where $x$ is the logarithm of $y$ to the base $b$, and${\log _b}y$ is the power to which we have to raise $b$ to get $y$, we are expressing $x$ in terms of $y$.
 Now the given question can be solved by using properties of both logarithms and exponents and then simplify the given question until we get an expression in terms of 5.

Complete step by step solution:
We know that logarithm is the power to which a number must be raised in order to get some other number, and the base unit is the number being raised to a power, For example, the base ten logarithm of 1000 is 3, because ten raised to the power of two is 100:$\log 1000 = 2$ because${10^3} = 1000$. In general, you write log followed by the base number as a subscript. The most common logarithms are base 10 logarithms and natural logarithms; they have special notations. A base ten log is written as$\log $, and we use different base units but most common logarithms are base 10 logarithms.
Now given expression is ${\log _5}625$,
Convert logarithm i.e., 625 in terms of 5 by prime factorisation, we get,
$ \Rightarrow 625 = 5 \times 5 \times 5 \times 5 = {5^4}$,
Now rewriting the expression we get,
$ \Rightarrow {\log _5}625 = {\log _5}{\left( 5 \right)^4}$,
Now, using logarithms property ${\log _a}{x^n} = n{\log _a}x$, so here $a = 5$, $x = 5$, and $n = 4$, by substituting the values in the identity we get,
$ \Rightarrow {\log _5}625 = 4{\log _5}\left( 5 \right)$,
Now we know that ${\log _a}a = 1$, we get,
$ \Rightarrow {\log _5}625 = 4\left( 1 \right)$,
Further simplification we get,
$ \Rightarrow {\log _5}625 = 4$.

$\therefore $ The value of ${\log _5}625$ will be equal to 4.

Note: A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number, in these types of questions, we use logarithmic properties and formulas, and some of useful formulas are:
 ${\log _a}xy = {\log _a}x + {\log _a}y$,
${\log _a}{x^n} = n{\log _a}x$,
${\log _a}b = \dfrac{{{{\log }_e}b}}{{{{\log }_e}a}}$,
${\log _{\dfrac{1}{a}}}b = - {\log _a}b$,
${\log _a}a = 1$,
${\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b$.