Question

How do you solve \[\log 2+\log \left( 4x-1 \right)=3\]?

Answer 1

Hint: This type of problem is based on the concept of logarithm. First, we have to consider the given equation and then use the property of logarithm, that is, \[\log \left( ab \right)=\log a+\log b\] to simplify the left-hand side of the equation. Here, a=2 and b=(4x-1). Then, convert the right-hand side of the equation into a logarithmic term by multiplying log10 with 3(since log10=1). Now, using the power rule of logarithm, that is, \[n\log a=\log {{a}^{n}}\] we get log1000 in the R.H.S. And by taking an antilog on the sides of the obtained equation, we get the value of x.

Complete step by step answer:
According to the question, we are asked to solve \[\log 2+\log \left( 4x-1 \right)=3\].
 We have been given the equation is\[\log 2+\log \left( 4x-1 \right)=3\]. ---------(1)
Let us first consider the L.H.S.
We know that \[\log \left( ab \right)=\log a+\log b\]. Using this property of logarithm in L.H.S., we get
\[\log 2+\log \left( 4x-1 \right)=\log 2\left( 4x-1 \right)\]
Now, let us use distributive property to simplify the log function, that is, \[a\left( b+c \right)=ab+ac\].
Therefore,
\[\log 2+\log \left( 4x-1 \right)=\log \left( 8x-2 \right)\]. -----------(2)
Now, consider R.H.S.
We know that log10=1.
\[\Rightarrow 3\log 10=3\]
Also, we know that \[n\log a=\log {{a}^{n}}\].
Using this power rule of logarithm in R.H.S., we get
\[3\log 10=\log {{\left( 10 \right)}^{3}}\]
\[\Rightarrow 3=\log {{\left( 10 \right)}^{3}}\] ------------(3)
Let us equate L.H.S. and R.H.S., we get
\[\log \left( 8x-2 \right)=\log {{\left( 10 \right)}^{3}}\] -----------(4)
Take antilog on both the sides of the equation (4)
Therefore, the log cancels out.
\[\Rightarrow 8x-2={{10}^{3}}\]
We know that \[{{10}^{3}}\]=1000. We get
\[8x-2=1000\]
Now, add 2 from both the sides of the equation.
\[\Rightarrow 8x-2+2=1000+2\]
On further simplification, we get
\[8x=1002\]
Divide both the sides of the equation by 8. We get,
\[\Rightarrow \dfrac{8x}{8}=\dfrac{1002}{8}\]
Cancelling out the common term, we get
\[x=\dfrac{501}{4}\]

Hence, the value of x for the given equation \[\log 2+\log \left( 4x-1 \right)=3\] is \[x=\dfrac{501}{4}\].

Note: Whenever we get such types of problems, we should make necessary calculations to the given equation and then find the value of x which is the required answer. We should be thorough with the properties of logarithm to solve this question. We can further simplify the answer by converting the final answer into decimal. That is, by dividing 501 by 4.
We get, \[x=\dfrac{501}{4}=125.25\].