Question

Given $2{\log _{10}}X + 1 = {\log _{10}}250$. Find $X$ and ${\log _{10}}2X$.

Answer 1

Hint- In order to solve this problem we will use the basic properties of logarithmic so by using this property we will make the equation in terms of $X$ and further by solving it we will get the value of $X$.

Complete step-by-step answer:
Given equation is $2{\log _{10}}X + 1 = {\log _{10}}250$
As we know the basic property of logarithmic
$alogb = log{b^a}$
By applying this property in above equation we have
${\log _{10}}{X^2} + 1 = {\log _{10}}250$
We know that $lo{g_{10}}10{\text{ }} = {\text{ }}1$, so we can write it
$b,$
Again using the properties of logarithmic as
$log{\text{ }}c{\text{ }} + {\text{ }}log{\text{ }}d{\text{ }} = {\text{ }}log{\text{ }}cd$ and if l $log{\text{ }}a{\text{ }} = {\text{ }}log{\text{ }}b$ then $a = b$
So we get
${\log _{10}}(10{X^2}) = {\log _{10}}(250)$
By using above property we will remove logarithm from both sides

$$\Rightarrow 10{X^2} = 250 \\\ \Rightarrow {X^2} = 25 \\\ \Rightarrow X = \pm 5 \\\$$

So we get the value of $X$ as $5$ and $- 5$ but negative number can not be used therefore the value of ${\text{X }} = {\text{ }}5$
i) $X = 5$
ii) ${\log _{10}}2X$
Substitute the value of x in above equation we get

$$\\\ = {\log _{10}}(2 \times 5) \\\ = {\log _{10}}(10) \\\ = 1 \\\$$

Hence the required answer is $1$.

Note- In mathematics the logarithm is the inverse exponentiation function. That means the logarithm of a given number $X$ is the exponent that must be increased to another fixed number, the base $b,$in order to produce the number $X$.