Question

How do you evaluate \[{{\log }_{7}}\left( 343 \right)\]?

Answer 1

Hint: To evaluate the given logarithmic term, we will use some of the properties of the logarithms. The first property states that, \[\log {{a}^{m}}=m\log a\]. Here, \[a\And m\in \] Real numbers and a is positive. Another property we will use is \[{{\log }_{a}}a=1\]. The given logarithmic term can be evaluated using these properties of logarithm.

Complete answer:
The given expression is of the form \[{{\log }_{a}}b\], here a = 7, and b = 343, which means that the base of logarithm is 7, and the argument of logarithm is 343. We know that logarithm can be simplified if the argument is in product form. The argument 343 has one of the factors as 7. So, the argument of the logarithm can also be written as,
\[\Rightarrow {{\log }_{7}}\left( 343 \right)={{\log }_{7}}\left( 7\times 49 \right)\]
Here, 49 also has 7 as one of the factors. So, the above expression can be written as,
\[\Rightarrow {{\log }_{7}}\left( 7\times 49 \right)={{\log }_{7}}\left( 7\times 7\times 7 \right)\]
In the above expression, 7 is being multiplied 3 times. Hence, we can replace the argument with \[{{7}^{3}}\]. By doing this, the above expression can be written as,
\[\Rightarrow \operatorname{l}{{\log }_{7}}\left( 7\times 7\times 7 \right)={{\log }_{7}}\left( {{7}^{3}} \right)\]
Using the property of logarithm, \[\log {{a}^{m}}=m\log a\] in the above expression, we get
\[\Rightarrow {{\log }_{7}}\left( {{7}^{3}} \right)=3{{\log }_{7}}\left( 7 \right)\]
If the base of logarithm and the argument of logarithm are the same, then its value equals one. Hence, we can say that \[{{\log }_{7}}\left( 7 \right)=1\]. Using this in the above expression, we get
\[\Rightarrow 3{{\log }_{7}}\left( 7 \right)=3\times 1\]
Hence, the value of the given logarithmic expression \[{{\log }_{7}}\left( 343 \right)\] equals 3.

Note: The given question can also be solved by using a different logarithmic property, as follows
We know that, \[{{7}^{3}}=343\]. Taking cube root of both sides of this, we get
\[\Rightarrow 7={{\left( 343 \right)}^{\dfrac{1}{3}}}\]
The given expression to evaluate is \[{{\log }_{7}}\left( 343 \right)\]. From the above expression, we can substitute \[{{\left( 343 \right)}^{\dfrac{1}{3}}}\] for \[7\] at the base of the logarithm. By doing this we get
\[\Rightarrow {{\log }_{{{\left( 343 \right)}^{\dfrac{1}{3}}}}}\left( 343 \right)\]
We know the property of logarithm, that states \[{{\log }_{{{a}^{n}}}}b=\dfrac{1}{n}{{\log }_{a}}b\]. Using this property in the above logarithm. We get
\[\begin{align}
  & \Rightarrow {{\log }_{{{\left( 343 \right)}^{\dfrac{1}{3}}}}}\left( 343 \right)=\dfrac{1}{\dfrac{1}{3}}{{\log }_{343}}\left( 343 \right) \\
 & \Rightarrow 3{{\log }_{343}}\left( 343 \right)=3(1) \\
 & \Rightarrow 3 \\
\end{align}\]
We are getting the same answer from both methods.