Question

How do you solve $\log (4x - 1) = 5$?

Answer 1

First we write the right hand side of the equation in log terms i.e. using the concept that value of log of a constant value can be written using the identity of logarithm i.e. $m\log n = \log {n^m}$. Equate both sides of the equation when all values are inside log on both sides of the equation. Shift values to the required side and calculate the value of x.

Complete step-by-step answer:
We have to solve the equation $\log (4x - 1) = 5$ … (1)
We know that value of $\log (10) = 1$ … (2)
We can write $5 = 5 \times 1$
Substitute the value of 1 using logarithm formula in equation (2)
$\Rightarrow 5 = 5 \times \log 10$ … (3)
Now we know the property of logarithm that $m\log n = \log {n^m}$
Here we compare the equation (3) with the property, then $m = 5,n = 10$
Then equation (3) becomes
$\Rightarrow 5 = \log {10^5}$ … (3)
Substitute this value of 5 from equation (3) in equation (1)
$\Rightarrow \log (4x - 1) = \log \left( {{{10}^5}} \right)$
Since the function is same on both sides, we can equate the brackets
$\Rightarrow 4x - 1 = {10^5}$
Shift constant values to right hand side of the equation
$\Rightarrow 4x = {10^5} + 1$
Expand the term on right hand side of the equation
$\Rightarrow 4x = 100000 + 1$
Add the constant values on right hand side of the equation
$\Rightarrow 4x = 100001$
Divide both sides of the equation by 4
$\Rightarrow \dfrac{{4x}}{4} = \dfrac{{100001}}{4}$
Cancel same factors from numerator and denominator on left side of the equation
$\Rightarrow x = \dfrac{{100001}}{4}$
Divide the numerator by denominator on right hand side of the equation
$\Rightarrow x = 25000.25$
$\therefore$The solution of equation $\log (4x - 1) = 5$ is $x = 25000.25$

Note:
Many students make mistake of opening the left hand side using log division property i.e. $\log \left( {\dfrac{m}{n}} \right) = \log m - \log n$ and then put in the value of $\log 1 = 0$. Keep in mind to open this term in this order we have to have log of a value minus log of another value, here we have subtraction as complete under one log so we cannot apply this property.