Question

Solve the equation \[\text{lo}{{\text{g}}_{3}}\left( 4x-5 \right)=5\].

Answer 1

Hint: This question is from the topic of pre-calculus. In this question, we will find out the value of x. In solving this question, we will remove the term log by using the formula of logarithmic function. After that, we will solve the further question of getting the value of x.

Complete step-by-step solution:
Let us solve this question.
In this question, we have asked to find the value of x from the given equation in the question.
The given equation in the question is
\[{{\log }_{3}}\left( 4x-5 \right)=5\]
Using the formula of logarithmic function that is: if \[{{\log }_{a}}b=c\], then we can write \[b={{a}^{c}}\].
So, we can write the equation \[{{\log }_{3}}\left( 4x-5 \right)=5\] as
\[\left( 4x-5 \right)={{3}^{5}}\]
As we know that \[{{3}^{5}}\] can also be written as \[3\times 3\times 3\times 3\times 3\], so we can write the above equation as
\[\Rightarrow \left( 4x-5 \right)=3\times 3\times 3\times 3\times 3\]
The above equation can also be written as
\[\Rightarrow \left( 4x-5 \right)=243\]
\[\Rightarrow 4x-5=243\]
Taking the number 5 to the right side of the equation, we can write the above equation as
\[\Rightarrow 4x=243+5\]
The above equation can also be written as
\[\Rightarrow 4x=248\]
Now, dividing 4 to the both side of equation, we can write the above equation as
\[\Rightarrow \dfrac{4x}{4}=\dfrac{248}{4}\]
The above equation can also be written as
\[\Rightarrow x=\dfrac{248}{4}\]
The above equation can also be written as
\[\Rightarrow x=62\]
Hence, we have solved the equation \[{{\log }_{3}}\left( 4x-5 \right)=5\] and got the value of x as 62.

Note: We should have a proper knowledge in the topic of pre-calculus to solve this type of question easily. We should know about the logarithmic functions. Remember the following formulas of logarithmic functions:
If \[{{\log }_{a}}b=c\], then we can write \[b={{a}^{c}}\]
\[{{\log }_{a}}1=0\]
\[\log \dfrac{b}{c}=\log b-\log c\]
\[\log \left( b\times c \right)=\log b+\log c\]
The above basic formulas should be kept remembered. They make our solving process easy.