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How do you simplify $ (\ln 3 - 2\ln 8) + \ln 16 $ ?

Last updated date: 25th Jul 2024
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Hint: Logarithms are expressed as the ways to figure out which exponents we need to multiply into the specific number. Here, using the property of logarithm the change of base, according to the power rule, the log of a power is equal to the power times the log of the base.
 $ \log {a^N} = N\log a $ along with the quotient and the product rule we will simply use the given expression.

Complete step-by-step answer:
Given expression: $ (\ln 3 - 2\ln 8) + \ln 16 $
Apply the power rule in the above expression, the log of a power is equal to the power times the log of the base.
 $ \log {a^N} = N\log a $
 $ = (\ln 3 - \ln {8^2}) + \ln 16 $
Simplify the above expression –
 $ = \ln 3 - \ln 64 + \ln 16 $
Apply, Quotient rule: $ {\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y $
 $ = \ln \left( {\dfrac{3}{{64}}} \right) + \ln 16 $
Apply, Product rule: $ {\log _a}xy = {\log _a}x + {\log _a}y $ in the above expression –
 $ = \ln \left( {\dfrac{3}{{64}} \times 16} \right) $
Remove common factors from the numerator and the denominator in the above expression.
 $ = \ln \left( {\dfrac{3}{4}} \right) $
This is the required solution.
So, the correct answer is “ $ \ln \left( {\dfrac{3}{4}} \right) $ ”.

Note: In other words, the logarithm can be defined as 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 for solution and it 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 $