Question

Evaluate the value of \[{\log _4}\left( 8 \right)\].

Answer 1

Hint:A logarithm can have any positive value as its base, but two log bases are more useful than the others. Logarithm is the inverse function to exponentiation. That means the logarithm of a given number “x” is the exponent to which another fixed number, the base “b”, must be raised, to produce that number “x”.

Complete step by step solution:
Rewriting as an equation we have,
\[{\log _4}\left( 8 \right) = x\]
If “x” and “b” are positive real numbers and “b” does not equal\[1\], then \[{\log _b}\left( x \right) =
y\]is equivalent to\[{b^y} = x\].
Hence we can write,
\[4x = 8\]
Now expressions in the equation that all have equal bases are created
\[ \Rightarrow {\left( {{2^2}} \right)^x} = {2^3}\]
This can be rewritten as,
\[ \Rightarrow {\left( 2 \right)^{2x}} = {2^3}\]
Since, the bases are the same, and then two expressions are only equal if the exponents are also equal.
Therefore we have,
\[ \Rightarrow 2x = 3\]
Now we will solve for\[x\],
\[ \Rightarrow x = \dfrac{3}{2}\]
The variable \[x\]is equal to \[\dfrac{3}{2}\].
The result can be shown in multiple forms.
In exact form\[x = \dfrac{3}{2}\],
In decimal form \[x = 1.5\] and
Mixed number \[x = 1\dfrac{1}{2}\]

Note: The logarithm is the inverse function to exponentiation. That means the logarithm of a given number “x” is the exponent to which another fixed number, the base “b”, must be raised, to produce that number “x”. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication. It is important to create equal bases for easy calculations since logarithmic scales reduce wide-ranging quantities to tiny scopes. Logarithm is a power to which a number must be raised in order to get some other number.