Question

How do you solve $\log (10 - 4x) = \log (10 - 3x)$ ?

Answer 1

Here we need to solve the above equation, for that we can apply the equality property of logarithm. This means that here, logarithms can be removed and we can equate the inside functions of the logarithms and then simplify that further.

Formula used:
Equality rule: If ${\log _a}x = {\log _a}y$ , then $x = y$ .

Complete step-by-step answer:
Logarithm is used to make complicated functions or calculations easy. The logarithm function has certain laws and properties, which can be used to make the simplifying process much simpler.
Here, we are using the equality property of logarithm which means that if it is given that two log functions with the same base are equal, then the log of the two functions can be removed and we can equate the remaining functions.
As we need to solve the equations, $\log (10 - 4x) = \log (10 - 3x)$
Removing the log on both sides by using the equality rule mentioned above, we get
$\Rightarrow 10 - 4x = 10 - 3x$, we can solve this by using simple algebra,
Adding $3\;x$ on both sides, we get
$\Rightarrow 10 - 4x + 3x = 10 - 3x + 3x$
$\Rightarrow 10 - x = 10$
Subtracting $\;10$ on both sides,
$\Rightarrow 10 - x - 10 = 10 - 10$
$\Rightarrow - x = 0$
Dividing by $- 1$ into both sides,
$\Rightarrow x = 0$ .

Hence, $x = 0$ is the required solution to the equation $\log (10 - 4x) = \log (10 - 3x)$

Additional information: Two systems of logarithms are generally used, which are, Common Logarithms: In this system, the base is always taken as $\;10$ . Natural Logarithms: In this system, the base is taken as $e$ , where $e$ is an irrational number lying between $2$ and $3$ .

Note:
Logarithm properties are very helpful in solving complicated exponential problems also. ‘Log’ is the abbreviated form of a logarithm. Product rule, power rule, quotient rule, etc. are the other laws and properties that can be used for the easier simplification process.