Question

How do you condense \[2\ln 4 - \ln 2?\]

Answer 1

Hint: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\[a\ln b = \ln b\].