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Question: How do you solve \({\log _3}x = 4\)?...

How do you solve log⁡3x=4{\log _3}x = 4?

Explanation

Solution

Use logarithm property xn=y{x^n} = y to get the answer
To solve this type of problem, we will have to use special logarithmic properties otherwise we won't be able to solve them. This is because operations in log are not done like the conventional way. So, in order to solve this question, we will use the property log⁡xy=n→xn=y{\log _x}y = n \to {x^n} = yto solve. Doing this we will get 34=x{3^4} = x which will in the end give x=81x = 81as the answer.

Complete step by step solution:
The given question we have is log⁡3x=4{\log _3}x = 4
To solve this problem, we will use a special property of logarithm, which states that:-
If there exists a log with base x and argument y. And if the value of that log y base x is equal to another number which may be a constant or variable lets say “n”. Then this particular equation can be written as x raised to the power n equals to y. If we represent it mathematically, it will look like:-
log⁡xy=n →xn=y  {\log _x}y = n \\\ \to {x^n} = y \\\
Both the steps are the same and you can use any one of it anytime.
So therefore, when we use this step on our given equation, we will get:-
log⁡3x=4 →34=x  {\log _3}x = 4 \\\ \to {3^4} = x \\\
Now, we know that.
34=3×3×3×3{3^4} = 3 \times 3 \times 3 \times 3
Which equals to 81
Therefore, x=81x = 81
And this is our solution for the question.

Note: Please remember the base of the log is always the base of the number which is raised to the power. The argument of the log is the value which is equal to the raised value of the base. So, in any case, don’t confuse yourself by exchanging the positions of base and argument. In that case you will get a wrong answer.