Question

How do you condense $3\ln 3 + \ln 9$?

Answer 1

Hint:In order to determine the condensed or we can say simplified form of the above question, rewrite the expression using the property of logarithm $n\log m = \log {m^n}$by taking $n = 3\,$and $m = 3$ then combine both of the logarithmic values into single logarithm by using the identity of addition of logarithm which says ${\log _b}(m) + {\log _b}(n) = {\log _b}(mn)$.By solving this you’ll get your desired condensed form.

Complete step by step solution:
We are Given an expression $3\ln 3 + \ln 9$ So to condense or simplify the expression we’ll be using some of the properties of logarithm.

$ = 3\ln 3 + \ln 9$
$3\ln 3$can be written as $\ln {3^3}$using the property of logarithm $n\log m = \log {m^n}$where $n = 3\,$and $m = 3$.

Replacing $3\ln 3$with in the expression $\ln {3^3}$
$ = \ln {3^3} + \ln 9$

As we know that any addition of two logarithmic values can be expressed as${\log _b}(m) + {\log_b}(n) = {\log _b}(mn)$here, m and n are ${3^3}$and $9$respectively

Now our equation becomes
$
= \ln \left( {{3^3} \times 9} \right) \\
= \ln \left( {27 \times 9} \right) \\
= \ln \left( {243} \right) \\
$
Therefore, the condensed form of the expression $3\ln 3 + \ln 9$is equal to $\ln \left( {243} \right)$.

Additional Information:
1.Value of constant ‘e’ is equal to $2.71828$.

2.A logarithm is basically the reverse of a power or we can say when we calculate a logarithm of any number, we actually undo an exponentiation.

3.Any multiplication inside the logarithm can be transformed into addition of two separate logarithm values.
${\log _b}(mn) = {\log _b}(m) + {\log _b}(n)$

4. Any division inside the logarithm can be transformed into subtraction of two separate logarithm values.
${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}(m) - {\log _b}(n)$

5. Any exponent value on anything inside the logarithm can be transformed and moved out of the logarithm as a multiplier and vice versa.
$n\log m = \log {m^n}$

Note:1.Don’t forget to cross-check your answer at least once.
2.$\ln $ is known as the “natural log” which is having base $e$.