Question

How do you solve $ 4{\log _5}(x + 1) = 4.8 $ ?

Answer 1

Hint: Take the given expression and frame the equation in the terms of the logarithm function and then apply the law of the logarithm to frame equation in terms of the “x” and then simplify for the resultant required value for it.

Complete step by step solution:
Take the given expression: $ 4{\log _5}(x + 1) = 4.8 $
Term multiplicative on one end if moved to the opposite side then it goes to the denominator.
 $ {\log _5}(x + 1) = \dfrac{{4.8}}{4} $
Remove the decimal point on the right hand side of the equation and place ten below it since we have only one digit after the decimal point.
 $ {\log _5}(x + 1) = \dfrac{{48}}{{4 \times 10}} $
Find factors on the right hand side of the equation in the numerator part.
 $ {\log _5}(x + 1) = \dfrac{{4 \times 12}}{{4 \times 10}} $
Common factors from the numerator and the denominator cancel each other. Therefore, remove from the numerator and the denominator.
 $ {\log _5}(x + 1) = \dfrac{{12}}{{10}} $
Again, convert the fraction on the right hand side of the equation in the decimal point system.
 $ {\log _5}(x + 1) = 1.2 $
Using the law of logarithm which states that if $ {\log _a}N = x $ then $ {a^x} = N $
 $ \Rightarrow {(5)^{1.2}} = x + 1 $
The above equation can be re-written as: $ x + 1 = {(5)^{1.2}} $
Simplify the above equation-
 $ \Rightarrow x + 1 = 6.898 $
Make “x” the subject, and move other terms on the opposite side. When you move any term from one side to another, the sign of the term also changes.
 $ \Rightarrow x = 6.898 - 1 $
Simplify the above equation-
 $ \Rightarrow x = 5.898 $
This is the required solution.
So, the correct answer is “x = 5.898 ”.

Note: Know the difference between ln and log and apply its properties accordingly. Logarithms are the ways to figure out which exponents we need to multiply into the specific number. In other words, the logarithm is the power to which the number must be raised in order to get some other. Always remember the standard properties of the logarithm.... Product rule, quotient rule and the power rule. The basic logarithm properties are most important and the solution solely depends on it, so remember and understand its application properly. Be good in multiples and know its concepts and apply them accordingly.