Question

Find the value of the logarithmic expression given as: \[\text{lo}{{\text{g}}_{3}}\text{9+lo}{{\text{g}}_{5}}\text{25+lo}{{\text{g}}_{2}}8\]. Choose the correct option:
A. 4
B. 5
C. 6
D. 7

Answer 1

Hint: We have to apply the logarithmic rule \[{{\log }_{a}}\left( {{x}^{b}} \right)=b\times {{\log }_{a}}\left( x \right)\] and the exponents rule, for example: \[9={{3}^{2}}\]. Also, we have to apply the logarithmic rule \[{{\log }_{a}}\left( a \right)=1\]. Simplify each term and find the final value by adding all.

Complete step-by-step answer:
bIn the question, we have to find the value of the logarithmic expression given as: \[\text{lo}{{\text{g}}_{3}}\text{9+lo}{{\text{g}}_{5}}\text{25+lo}{{\text{g}}_{2}}8\]
So here, we have to apply the logarithmic rules and the exponents rule. The logarithmic rule that is will be applied here is \[{{\log }_{a}}\left( {{x}^{b}} \right)=b\times {{\log }_{a}}\left( x \right)\]. Now, we have to understand the values of a, b and x from the given terms of the logarithmic expressions \[\text{lo}{{\text{g}}_{3}}\text{9+lo}{{\text{g}}_{5}}\text{25+lo}{{\text{g}}_{2}}8\].
Now we will first take the first term which is \[\text{lo}{{\text{g}}_{3}}\text{9}\], in this we will apply the rule \[{{\log }_{a}}\left( {{x}^{b}} \right)=b\times {{\log }_{a}}\left( x \right)\], to obtain \[\text{lo}{{\text{g}}_{3}}\text{9}={{\log }_{3}}\left( {{3}^{2}} \right)=2{{\log }_{3}}\left( 3 \right)\]. Now we will apply the logarithmic rule \[{{\log }_{a}}\left( a \right)=1\], to get:
\[\begin{align}
  & \Rightarrow \text{lo}{{\text{g}}_{3}}\text{9}={{\log }_{3}}\left( {{3}^{2}} \right) \\
 & \Rightarrow \text{lo}{{\text{g}}_{3}}\text{9}=2{{\log }_{3}}\left( 3 \right) \\
 & \Rightarrow \text{lo}{{\text{g}}_{3}}\text{9}=2\times \;1 \\
 & \Rightarrow \text{lo}{{\text{g}}_{3}}\text{9}=2 \\
\end{align}\]
Next, we will take the next term \[\text{lo}{{\text{g}}_{5}}\text{25}\], and find the value, as follows:
\[\begin{align}
  & \Rightarrow {{\log }_{5}}\left( 25 \right)={{\log }_{5}}\left( {{5}^{2}} \right) \\
 & \Rightarrow {{\log }_{5}}\left( 25 \right)=2{{\log }_{5}}\left( 5 \right) \\
 & \Rightarrow {{\log }_{5}}\left( 25 \right)=2\times \;1\,\,\,\,\,\,\,\,\,\,\because {{\log }_{5}}\left( 5 \right)=1 \\
 & \Rightarrow {{\log }_{5}}\left( 25 \right)=2 \\
\end{align}\]
Also, we will find the value of the third term \[\text{lo}{{\text{g}}_{2}}8\]n using the same set of logarithmic rules. So, we get:
\[\begin{align}
  & \Rightarrow \text{lo}{{\text{g}}_{2}}8={{\log }_{2}}\left( {{2}^{3}} \right) \\
 & \Rightarrow \text{lo}{{\text{g}}_{2}}8=3{{\log }_{2}}\left( 2 \right) \\
 & \Rightarrow \text{lo}{{\text{g}}_{2}}8=3\times 1\,\,\,\,\,\,\,\,\,\,\because {{\log }_{2}}\left( 2 \right)=1 \\
 & \Rightarrow \text{lo}{{\text{g}}_{2}}8=3 \\
\end{align}\]
So, finally, the value of the logarithmic expression given as:
 \[\begin{align}
  & \Rightarrow \text{lo}{{\text{g}}_{3}}\text{9+lo}{{\text{g}}_{5}}\text{25+lo}{{\text{g}}_{2}}8=2+2+3 \\
 & \Rightarrow \text{lo}{{\text{g}}_{3}}\text{9+lo}{{\text{g}}_{5}}\text{25+lo}{{\text{g}}_{2}}8=7 \\
\end{align}\]
Hence, we have obtained the final value as 7. So, the correct answer is option D.

Note: We have to be careful while simplifying the logarithmic expression and applying the rule \[{{\log }_{a}}\left( {{x}^{b}} \right)=b\times {{\log }_{a}}\left( x \right)\]. The common mistake can be when we are solving \[\text{lo}{{\text{g}}_{3}}\text{9}\], \[\text{lo}{{\text{g}}_{2}}8\] and \[\text{lo}{{\text{g}}_{5}}\text{25}\]. So, before applying the formula we have to first compare and find the values of a, b and x in the formula above.