Question

Solve for $x$
${{\log }_{3}}x-{{\log }_{3}}2=1$

Answer 1

In this question we have to solve the given term ${{\log }_{3}}x-{{\log }_{3}}2=1$ for $x$ hence we can see that the base is same so we can apply the quotient rule law that is ${{\log }_{a}}M-{{\log }_{a}}N={{\log }_{a}}\dfrac{M}{N}$ and $1$ on the right hand side can be written as ${{\log }_{3}}3$ , then we will solve the equation to find the value of $x$

Complete step-by-step solution:
Even before the discovery of calculus, mathematicians employed logarithms to convert division and multiplication problems into subtraction and addition issues. To get a given number in logarithm, the power is raised to some number, which is usually a base number. You may write any exponential function in logarithmic form because logarithmic functions are the inverse of exponential functions. All logarithmic functions are rewritten in exponential form as well. Logarithms are useful in managing numbers of a much more manageable scale when working with very huge numbers.
The concept of logarithms was first developed in the ${{17}^{th}}$ century by John Napier. Many scientists, navigators, engineers, and others then utilized it to conduct numerous computations, making it simple. To put it another way, logarithms are the inverse of exponentiation.
The power to which a number must be raised in order to obtain additional values is defined as a logarithm. It is the most practical method of expressing enormous numbers. Multiplication and division of logarithms can also be stated in the form of logarithms of addition and subtraction, thanks to a number of important features of logarithms.
Logarithm is of two types that is common logarithm and natural logarithm. Where the base $10$ logarithms are also known as the common logarithm. It is written as log $10$ or just log and the base $e$ logarithm is the natural logarithm. The natural logarithm is denoted by the letters ln or log $e$.
Now according to the question we need to find the value of $x$ of the given term:
$\Rightarrow {{\log }_{3}}x-{{\log }_{3}}2=1$
As we know that ${{\log }_{e}}a=M$ is equal to ${{e}^{M}}=a$
And $1$ on the right hand side can be written as ${{\log }_{3}}3$
$\Rightarrow {{\log }_{3}}x-{{\log }_{3}}2={{\log }_{3}}3$
On left hand side apply the quotient rule law that is ${{\log }_{a}}M-{{\log }_{a}}N={{\log }_{a}}\dfrac{M}{N}$
$\Rightarrow {{\log }_{3}}\left( \dfrac{x}{2} \right)={{\log }_{3}}3$
$\Rightarrow \left( \dfrac{x}{2} \right)=3$
$\Rightarrow x=3\times 2$
$\Rightarrow x=6$

Note: People utilized logarithm tables in books to multiply and divide before calculators. A slide rule, an instrument with logarithms written on it, has the same information as a logarithm table. Adding logarithms is the same as multiplying and subtracting logarithms is the same as dividing, logarithms can make multiplication and division of huge numbers easier.