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 $X$.