Question

How do you simplify $\log {{100}^{x}}$ ?

Answer 1

Hint: To simplify $\log {{100}^{x}}$, we are going to use the following property of exponents in logarithm: $\log {{a}^{b}}=b\log a$. Then we will write 100 in $\log {{100}^{x}}$ as ${{10}^{2}}$. The base of the power has not been mentioned so in general the base of the logarithm is taken as 10 so we are going to use the property of logarithm that ${{\log }_{a}}a=1$.

Complete step by step answer:
The logarithmic expression which we have to simplify is as follows:
$\log {{100}^{x}}$
To simplify the above expression, we will use the following property of logarithm which states that:
$\log {{a}^{b}}=b\log a$
Using the above property, we are going to take the exponent of 100 out from the logarithm and put it before log and in multiplication with log.
$x\log 100$
Now, as you can see that if we write 100 as ${{10}^{2}}$ then again we can use the above exponent property of logarithm so writing 100 as ${{10}^{2}}$ in the above expression we get,
$x\log {{\left( 10 \right)}^{2}}$
Using the above exponent property of logarithm we can take the power of 2 out from logarithm and then write that power of 2 in multiplication with x.
$2x\log 10$
We know that, generally, the base of logarithm is 10 so we can use the following property of logarithm to simplify the above expression.
${{\log }_{a}}a=1$
In $2x\log 10$, “a” is 10 here so we are writing ${{\log }_{10}}10=1$ in $2x\log 10$.
$2x\left( 1 \right)$
Multiplying 1 with $2x$ we get,
$2x$

Hence, we have simplified the given expression to $2x$.

Note: To crack the above problem, you must know the following properties of logarithm:
$\log {{a}^{b}}=b\log a$
${{\log }_{a}}a=1$
Failure of knowledge of any of the above properties will make you handicap in this problem so make sure you have a good understanding of them.
One more thing, in the above problem base of the logarithm has not been mentioned so we have taken it as 10 because first of all, it will ease out the calculation and secondly, in general the base of logarithm is 10 because when base of logarithm is “e” then logarithm is written as “ln”.