Question

How do you evaluate \[{\log _5}1\]?

Answer 1

Hint: Here we will first assume the value of the given logarithmic expression to be any variable. Then we will use the properties of the logarithmic function and convert it into an exponential function. We will simplify the given equation further to get the required answer.

Complete step by step solution:
Here we need to find the value of the given logarithmic expression. The given logarithmic expression is \[{\log _5}1\].
Let the value of \[{\log _5}1\] be \[x\].
We can write it as
\[{\log _5}1 = x\] ………….. \[\left( 1 \right)\]
We will use the properties of the logarithmic function here.
According to the property of the logarithmic expression when \[{\log _a}b = x\], then \[b = {a^x}\], where, \[a\], \[b\] and \[x\] are the real numbers.
Using this property of logarithmic function, we get
\[ \Rightarrow 1 = {5^x}\]
This is possible only when \[x = 0\].
Now, we will substitute the value of \[x\] in equation \[\left( 1 \right)\]. Therefore, we get

\[{\log _5}1 = 0\]

Hence, this is the required value of the given logarithmic expression.

Additional information:
We know that the log function is different from the ln function. The difference between ln and log is that ln is represented for the base \[e\] but the log is denoted for base 10. For example, log with base 4 is represented as \[{\log _4}\] and log with base \[e\] is represented by \[{\log _e} = \ln \].

Note:
We know that the logarithmic function is defined as the function inverse of the exponential function. We need to remember the basic property of the logarithmic function. When we add two logarithmic functions, then the result of the addition will be equal to the logarithm with their base as the product of the bases of the first two logarithms.