Question

How do you solve $\ln (2x - 5) = 3.78$ ?

Answer 1

Hint : Logarithms are the ways to figure out which exponents we need to multiply into the specific number. Log is defined for the base $10$ and ln is denoted for the base e. “e” is an irrational and transcendental number which can be expressed as $e = 2.71828$ and uses the property and simplify for the required value.

Complete step-by-step answer :
Take the given expression: $\ln (2x - 5) = 3.78$
Use the definition of log to change it into an exponential form:
$(2x - 5) = {e^{3.78}}$
Make the required term “x” the subject. When you move any term from one side to another side then the sign of the term changes. Positive term changes to negative and vice-versa.
$2x = {e^{3.78}} + 5$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$x = \dfrac{{{e^{3.78}} + 5}}{2}$
Place $e = 2.71828$ and simplify –
$x = 24.41$
This is the required solution.
So, the correct answer is “ $x = 24.41$ ”.

Note : 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.
Also refer to the below properties and rules of the logarithm.
Product rule: ${\log _a}xy = {\log _a}x + {\log _a}y$
Quotient rule: ${\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y$
Power rule: ${\log _a}{x^n} = n{\log _a}x$
Base rule: ${\log _a}a = 1$