Question

How do you simplify the expression ${{\log }_{2}}256$?

Answer 1

Hint: Now we are given with a logarithmic expression with base 2. To find the value of the given expression we will use the definition of logarithm. We know that if ${{\log }_{a}}b=x$ then we have ${{a}^{x}}=b$ Hence we need to find what must be raised to 2 so that we get 256.

Complete step by step solution:
Now first let us understand the function log.
Now the function log is normally written as ${{\log }_{b}}x$ where b is called the base of log.
Now the base of log is a fixed number and x is the variable.
Now let us understand the meaning of the function.
Suppose we have an expression of the form ${{a}^{x}}=b$ then in logarithm we write it as ${{\log }_{a}}b=x$
Hence log gives the value to which based must be raised to obtain the given number.
Let us take an example to understand this,
Let us say we have ${{\log }_{10}}100$ then the value of this is 2 as ${{10}^{2}}=100$ .
Now let us understand three basic properties of logarithm.
The exponent law. ${{\log }_{a}}{{b}^{n}}=n{{\log }_{a}}b$ .
The multiplication law. ${{\log }_{a}}\left( bc \right)={{\log }_{a}}b+{{\log }_{a}}\left( c \right)$
The division law ${{\log }_{a}}\left( \dfrac{c}{b} \right)={{\log }_{a}}c-{{\log }_{a}}b$
Now we can use these properties to simplify the logarithmic equations.
Now consider the given expression ${{\log }_{2}}256$
We know that ${{2}^{8}}=256$ . Hence using the definition of logarithm we can say that ${{\log }_{2}}256=8$ .

Note:
Now note that the base of log can be any positive number other than 1. If there is no base written in a logarithmic expression then we assume the base to be 1. If the base of the log is constant e then the log is called the natural logarithm and is represented as ln.