Question

How do you condense $2\ln 4 - \ln 2?$

Answer 1

The given question describes the operation of addition/ subtraction/ multiplication/ division with the involvement of a natural algorithm. We need to know the basic formula for$\ln$ multiplication and $\ln$ division. We should try to compare the given equation with the basic$\ln$condition to make an easy calculation.

Complete step by step solution:
The given equation is shown below,
$2\ln 4 - \ln 2 = ?$

The above equation can also be written as,
$2\ln 4 = \ln 2 \to \left( 1 \right)$

We know that,
$a\ln b = \ln {b^a} \to \left( 2 \right)$

By comparing the LHS of the equation$\left( 1 \right)$with the equation$\left( 2 \right)$, we get
$2\ln 4$
$\downarrow$
$a\ln b$

So, we get,

$$a = 2 \\\ b = 4 \\\$$

So, we get

$$2\ln 4 = a\ln b \\\ \left( 2 \right) \to a\ln b = \ln {b^a} \\\ 2\ln 4 = \ln {4^2} \to \left( 3 \right) \\\$$

By substituting the equation $\left( 3 \right)$in the equation$\left( 1 \right)$we get,

$$2\ln 4 = \ln 2 \\\ \ln {4^2} = \ln 2 \\\$$

So, the above equation can also be written as
$\ln {4^2} - \ln 2 = 0 \to \left( 4 \right)$

We know that,
$\ln p - \ln q = \ln \left( {\dfrac{p}{q}} \right) \to \left( 5 \right)$

By comparing the equation $\left( 4 \right)$and$\left( 5 \right)$ we get,

$$\left( 5 \right) \to \ln p - \ln q = \ln \left( {\dfrac{p}{q}} \right) \\\ \left( 4 \right) \to \ln {4^2} - \ln 2 = 0 \\\$$

So we get,

$$p = {4^2} = 16 \\\ q = 2 \\\$$

So, the value of$\ln {4^2} - \ln 2$becomes,

$$\ln {4^2} - \ln 2 = \ln \left( {\dfrac{{{4^2}}}{2}} \right) \\\ \ln {4^2} - \ln 2 = \ln \left( {\dfrac{{16}}{2}} \right) \\\ \ln {4^2} - \ln 2 = \ln \left( 8 \right) \\\$$

By using a calculator, we get
$\ln (8) = 2.0794$

So, the final answer is,
$2\ln \left( 4 \right) - \ln \left( 2 \right) = \ln \left( 8 \right) = 2.0794$

Note: This type of question involves the operation of addition/ subtraction/ multiplication/ division with the involvement of a natural algorithm. We should remember the basic conditions in the natural algorithm calculations.
By using the scientific calculator we can convert the final answer into a decimal number. Also, note the following things,

1. $\ln \dfrac{a}{b} = \ln \left( a \right) - \ln \left( b \right)$, In natural algorithm
   calculations, if two terms are involved in$\left( {\ln } \right)$the division, we can separate
   the two terms with the involvement of$\left( {\ln } \right)$subtraction as shown in the
   mentioned formula.
2. $\ln \left( {a \cdot b} \right) = \ln \left( a \right) + \ln \left( b \right)$, In natural algorithm calculations, if two terms are involved in$\left( {\ln } \right)$ multiplication, we can separate the two terms with the involvement of$\left( {\ln } \right)$addition as shown in the mentioned formula.
3. $a\ln b = \ln {b^a}$, if the value of$a$is$1$ the mentioned formula can be written
   as$a\ln b = \ln b$.