Question

Find the value of
\[\begin{align}
  & i){{\log }_{2}}32 \\
 & ii){{\log }_{5}}3\sqrt{5} \\
 & iii){{\log }_{\sqrt{2}}}64 \\
 & iv){{\log }_{10}}100000 \\
\end{align}\]

Answer 1

Hint: The above question uses the some elementary concepts of logarithm.
${{\log }_{a}}N=x$ read as log of N to the base a if a = e, we write $\ln N\Rightarrow {{\log }_{e}}N$ (Natural log).
The three major properties of logarithm is used in this question:
\[\begin{align}
  & \to {{\log }_{a}}a=1 \\
 & \to {{\log }_{a}}{{x}^{p}}=p{{\log }_{a}}x;x\text{ }>\text{ }0 \\
 & \to {{\log }_{{{a}^{q}}}}x=\dfrac{1}{q}{{\log }_{a}}x;x\text{ }>\text{ }0 \\
\end{align}\]

Complete step-by-step solution:
Now, we have given data as:
\[\left( \text{i} \right){{\log }_{2}}32\]
Here, we can write $32={{2}^{5}}$ therefore, we get ${{\log }_{2}}{{2}^{5}}$
Now, according to property ${{\log }_{a}}{{x}^{p}}=p{{\log }_{a}}x$ here, \[{{\log }_{2}}{{2}^{5}}=5{{\log }_{2}}2\]
Now, we know property ${{\log }_{a}}a=1$ hence, \[5{{\log }_{2}}2=5\times 1=5\]
Hence, answer of (i) is \[{{\log }_{2}}32=5\]
Now, coming to the second part of the question i.e. \[\left( ii \right){{\log }_{5}}3\sqrt{5}\]
We know that, \[3\sqrt{5}={{\left( 5 \right)}^{\dfrac{1}{3}}}\left[ \because n\sqrt{y}={{\left( y \right)}^{\dfrac{1}{n}}} \right]\]
Therefore, we get ${{\log }_{5}}{{\left( 5 \right)}^{\dfrac{1}{3}}}$
Now, apply the property ${{\log }_{a}}{{x}^{p}}=p{{\log }_{a}}x$
\[\Rightarrow {{\log }_{5}}{{\left( 5 \right)}^{\dfrac{1}{3}}}=\dfrac{1}{3}{{\log }_{5}}5\]
By applying property ${{\log }_{a}}a=1$ we get:
\[\Rightarrow \dfrac{1}{3}{{\log }_{5}}5=\dfrac{1}{3}\times 1=\dfrac{1}{3}\]
Hence, answer of (ii) is \[{{\log }_{5}}3\sqrt{5}=\dfrac{1}{3}\]
Now, coming to the third part of the question, i.e. \[\left( iii \right){{\log }_{\sqrt{2}}}64\]
Now, this part uses the different properties as compared to the previous two parts.
Here, we can write \[\begin{align}
  & \sqrt{2}={{\left( 2 \right)}^{\dfrac{1}{2}}} \\
 & \therefore {{\log }_{{{\left( 2 \right)}^{\dfrac{1}{2}}}}}64 \\
\end{align}\]
Now, using property ${{\log }_{{{a}^{q}}}}x=\dfrac{1}{q}{{\log }_{a}}x$
We have, \[{{\log }_{{{\left( 2 \right)}^{\dfrac{1}{2}}}}}64=2{{\log }_{2}}64\]
Now, 64 can be written as ${{2}^{6}}$
\[\therefore 2{{\log }_{2}}{{\left( 2 \right)}^{6}}\]
Now, again using property ${{\log }_{a}}{{x}^{p}}=p{{\log }_{a}}x$ we have $2\times 6{{\log }_{2}}\left( 2 \right)$
Finally, by applying property ${{\log }_{a}}a=1$ we get:
\[\Rightarrow 2\times 6\times {{\log }_{2}}\left( 2 \right)=2\times 6\times 1=12\]
Hence, answer of (iii) is \[{{\log }_{\sqrt{2}}}64=12\]
Now, coming to the fourth part of the question i.e. \[\left( iv \right){{\log }_{10}}100000\]
Here, we can write 100000 as ${{10}^{5}}$
\[\therefore {{\log }_{10}}{{\left( 10 \right)}^{5}}\]
Using property ${{\log }_{a}}{{x}^{p}}=p{{\log }_{a}}x$ we have:
\[\Rightarrow {{\log }_{10}}{{\left( 10 \right)}^{5}}=5{{\log }_{10}}10\]
Finally, using property ${{\log }_{a}}a=1$ we have:
\[\Rightarrow 5{{\log }_{10}}10=5\times 1=5\]
Hence, answer of (iv) is \[{{\log }_{10}}100000=5\]

Note: The most common mistake that may occur in logarithm is "the expression ${{\log }_{a}}{{x}^{p}}$ is written as ${{\left( {{\log }_{a}}x \right)}^{p}}$". We can’t write ${{\log }_{a}}{{x}^{p}}={{\left( {{\log }_{a}}x \right)}^{p}}$. The other important thing is necessary conditions.
Like we have ${{\log }_{a}}N=x$ then $N\text{ }>\text{ }0,a\text{ }>\text{ }0,a\ne 1$
Some general points like we have been given logN (where base of log is not given in the question) then we will take base as 'e'. Therefore, ${{\log }_{e}}N=\ln N$