Question

How do you evaluate $\log 900-\log 9$?

Answer 1

Hint: Check out all the laws of logarithms. The most important law of logarithms used here is the powers of logarithms and the subtraction of the logarithms algorithm. First, convert the first term into power form and then use the subtraction law to subtract both the terms to get a single term.

Complete step by step solution:
Assuming that the bases of the logarithmic terms are the same and that when bases are not mentioned, we assume the base as $10\;$
The given expression is $\log 900-\log 9$.
Now, consider the first term which is $\log 900$
We can write $\log 900$ as $\log \left( 3\times 3\times 10\times 10 \right)$
Further writing it into powers form we get,
$\Rightarrow \log \left( {{3}^{2}}\times {{10}^{2}} \right)$
Since we know the Law of the addition of logarithms, if we have the expression, ${{\log }_{a}}\left( b \right)+{{\log }_{a}}\left( c \right)$ . It is equal to ${{\log }_{a}}\left( bc \right)$.
We can rewrite it as,
$\Rightarrow \log \left( {{3}^{2}} \right)+\log \left( {{10}^{2}} \right)$
Since we know that $\log \left( {{b}^{c}} \right)=c\log \left( b \right)$,
Now, write it in the power form.
$\Rightarrow 2\log \left( 3 \right)+2\log \left( 10 \right)$
Now, rewrite all the terms together.
$\Rightarrow 2\left[ \log \left( 3 \right)+\log \left( 10 \right) \right]-\log 9$
We can write $\log 9$ as $\log {{3}^{2}}$ which can be further written in power form as,
$\Rightarrow 2\log 3$
Now our expression is,
$\Rightarrow 2\left[ \log \left( 3 \right)+\log \left( 10 \right) \right]-2\log 3$
$\Rightarrow 2\left[ \log 3+\log \left( 10 \right)-\log 3 \right]$
On evaluation we get,
$\Rightarrow 2\left[ \log \left( 10 \right) \right]$
Since we assumed the base as $10\;$ ,
On substitution we get,
$\Rightarrow 2\left[ {{\log }_{10}}10 \right]$
If the base and the logarithm value is the same, they cancel out to get $1$
$\Rightarrow {{\log }_{10}}10=1$
$\Rightarrow 2{{\log }_{10}}10=2$
Hence the expression $\log 900-\log 9=2$

Note: One should ensure that always the base of the given logarithms is the same before evaluating the expression using the laws of the logarithms. Sometimes logarithm is written without a base. In this case, we must assume that the base is 10. It is called the “common logarithm”.