Question

How do you simplify \[\log (8) + \log (18)\] ?

Answer 1

Hint: In this question, we have to use properties of logarithmic function. Here we have to use several properties like $\log {a^n} = n\log a$ and $\log \left( {a \times b} \right) = \log a + \log b$. We have to express our given question in a form where these identities can be applied easily. After simplification, we have to return the answer in terms of logarithmic function.

Complete step-by-step answer:
In the above question, first we will convert the given two terms in the most simplified form and then we will add them.
So, first we will convert $\log 8$ into its simplified form.
Now we have to simplify the logarithmic expressions $\log 8$using basic logarithmic identity of power rule.
$\log \left( 8 \right) = \log {\left( 2 \right)^3}$
Therefore, we will use the identity $\log {a^n} = n\log a$ here
$\log \left( 8 \right) = 3\log \left( 2 \right).....................\left( 1 \right)$
Now we will convert $\log 18$ into its most simplified form.
$\log \left( {18} \right) = \log \left( {2 \times 9} \right)$
Now, we will use identity $\log \left( {a \times b} \right) = \log a + \log b$
Therefore, we can also write it as
$\log \left( {18} \right) = \log \left( 2 \right) + \log \left( 9 \right)$
Now we will again use the identity $\log {a^n} = n\log a$.
$\log \left( {18} \right) = \log \left( 2 \right) + \log {\left( 3 \right)^2}$
$\log \left( {18} \right) = \log \left( 2 \right) + 2\log \left( 3 \right)....................\left( 2 \right)$
Now add equation $\left( 1 \right)$and $\left( 2 \right)$
\[\log (8) + \log (18) = 3\log \left( 2 \right) + \log \left( 2 \right) + 2\log \left( 3 \right)\]
\[\log (8) + \log (18) = 4\log \left( 2 \right) + 2\log \left( 3 \right)\]

Note: Logarithmic properties useful for similar questions of derivatives are listed below:
\[1.ln(PQ) = ln(P) + ln(Q)\]
\[2.ln\left( {\dfrac{P}{Q}} \right) = ln(P) - ln(Q)\]
\[3.ln{(P)^q} = q \times ln(P)\]
Equations $'1'$ , $'2'$ and $'3'$ are called Product Rule, Quotient Rule and Power Rule respectively. Also, another basic identity which is necessary to solve logarithmic equations is \[ln1 = 0\] where ‘a’ is any real number. Common logarithmic functions are log functions with base $10$, and natural logarithmic functions are log functions with base ‘e’. Natural logarithmic functions can be represented by \[lnx\,or\,lo{g_e}x\].