Question

How do you solve \[\log \left( x \right) + \log \left( {x + 1} \right) = \log \left( {12}
\right)\]?

Answer 1

Hint:In the given question, we need to find the value of x as per the equation \[\log \left( x \right) + \log \left( {x + 1} \right) = \log \left( {12} \right)\]. Apply the law of logarithms to solve this equation which is\[\log a + \log b = \log ab\],further apply log properties to solve this equation.

Complete step by step answer:
Let us write the given equation
\[\log \left( x \right) + \log \left( {x + 1} \right) = \log \left( {12} \right)\]
Here we can see in the equation that all the terms are with respect to log function
\[1 = \log \left( x \right)\]
If the base is 10, then we can write the above term as
\[{10^1} = x\]
This implies x=10
Hence,
\[1 = \log 10\]
Now let us rewrite the given equation with respect to the terms implied
\[\log x + \log x + \log 10 + \log 12\]
As the general rule of the logarithm is
\[\log a \cdot b \cdot c = \log a + \log b + \log c\]
Hence applying the general rule to the equation, we get
\[\log x + \log x + \log 10 = \log x \cdot x \cdot 10\]
Therefore, after simplifying we get
\[\log 10{x^2} = \log 12\]
This implies that if the logs are equal, then the numbers are equal
Hence, by basic definition of logarithms:
\[10{x^2} = 12\]
\[{x^2} = \dfrac{6}{5}\]
Therefore, the value of x is
\[x = \sqrt {\dfrac{6}{5}} \]
Additional information: Rules of Logarithms
The logarithm of a positive real number can be negative, zero or positive.
Logarithmic values of a given number are different for different bases.
Logarithms to the base a 10 are referred to as common logarithms. When a logarithm is written without a subscript base, we assume the base to be 10.
The logarithmic value of a negative number is imaginary and the logarithm of any positive number to the same base is equal to 1.
\[{a^1} = a \Rightarrow {\log _a}a = 1\]
The logarithm of 1 to any finite non-zero base is zero.
\[{a^0} = 1 \Rightarrow {\log _a}1 = 0\]
Formula used:
\[\log a + \log b = \log ab\]

Note: The key point to find the value of x in the given equation is that applying the formula \[\log a + \log b = \log ab\], when the equation consists of two variables and hence by applying the logarithmic properties, we can get the value of x