Question

Solve \[2\log x - \log y = 1\] where base of logarithm is \[10\].

Answer 1

Hint:In the above given question, we are given an equation containing the logarithmic functions written as \[2\log x - \log y = 1\] . We have to solve this above given equation, that means we have to write the equation in another form such that the equation becomes independent of the logarithms and contains only the variables and constants. In order to approach the solution, we have to use some logarithms to rewrite the given equation in the required form.

Complete step by step answer:
Given that, a logarithmic equation as,
\[ \Rightarrow 2\log x - \log y = 1\]
We have to find the solution of the above given logarithmic equation which is independent of logarithms. It is given that the base of the given logarithm is \[10\]. Now, since we have \[{\log _a}a = 1\] ,
Therefore, here we can write it as
\[ \Rightarrow {\log _{10}}10 = 1\]
Substituting this value in the given equation, that gives us
\[ \Rightarrow 2\log x - \log y = {\log _{10}}10\]

Now, since we know that \[\log {a^n} = n\log a\] , therefore we can write
\[ \Rightarrow \log {x^2} - \log y = {\log _{10}}10\]
Now, using the subtraction identity of logarithms, that is given by \[\log a - \log b = \log \dfrac{a}{b}\] ,
Hence, we can write the above equation as,
\[ \Rightarrow \log \dfrac{{{x^2}}}{y} = {\log _{10}}10\]
Now, since we have logarithmic signs in both the sides, i.e. in the LHS and RHS,
Therefore, cancelling logarithmic signs from both sides, we can write the above equation as,
\[ \Rightarrow \dfrac{{{x^2}}}{y} = 10\]
Now, multiplying both sides by \[y\] gives us the equation,
\[ \Rightarrow {x^2} = 10y\]
That is the required solution.

Hence, the solution of the equation \[2\log x - \log y = 1\] is given as \[{x^2} = 10y\].

Note:A logarithm is the power to which a number must be raised in order to get some other number. In other words, a logarithm is the exponent or power to which a base must be raised to yield a given number.
Mathematically, \[x\] is the logarithm of \[n\] to the base \[b\] if,
\[ \Rightarrow {b^x}\; = {\text{ }}n\]
Then we can write,
\[ \Rightarrow x{\text{ }} = {\text{ }}lo{g_b}\;n\]