Question

How do I use logarithm to solve for $x$ if ${{7} ^ {2x-1}} =316$?

Answer 1

Hint: Before solving the equation I want to give basic explanation on logarithm. Logarithm is the power to which the number must be raised in order to get some other number. In this equation we will use some basic rules of logarithm. First rule of logarithm is: $\ln {{\left( a \right)}^{b}}=b\ln \left( a \right)$ and the second rule of logarithm is:$\ln \left( a \right)\cdot \ln \left( b \right)=\ln \left( a \right)+\ln \left( b \right)$ and the third formula of logarithm is:$\dfrac{\ln \left( a \right)}{\ln \left( b \right)}=\ln \left( a \right)-\ln \left( b \right)$ . These are the three basic rules of logarithm.

Complete step by step solution:
Now the given equation is,
$\Rightarrow {{7} ^ {2x-1}} =316$
Now first apply natural log on both sides, we get,
$\Rightarrow \ln \left ({{7} ^ {2x-1}} \right)=\ln \left (316 \right)$
Here we can clearly see the LHS part, we will simplify LHS part by using the first rule of logarithm: $\ln {{\left (a \right)}^{b}}=b\ln \left (a \right)$, we get,
 $\Rightarrow \left (2x-1 \right)\ln \left (7 \right)=\ln \left (316 \right)$
Now we will use some basic algebraic skills to solve the above equation, first we will divide the whole equation with $\ln \left( 7 \right)$ on both sides, we get,
$\Rightarrow \dfrac{\left( 2x-1 \right)\ln \left( 7 \right)}{\ln \left( 7 \right)}=\dfrac{\ln \left( 316 \right)}{\ln \left( 7 \right)}$
Both $\ln \left (7 \right)$get cancel out in LHS part, we get,
$\Rightarrow 2x-1=\dfrac {\ln \left (316 \right)}{\ln \left (7 \right)}$
Now we will add $1$ on both side of equation, we get
$\Rightarrow 2x=\dfrac {\ln \left (316 \right)}{\ln \left (7 \right)}+1$
Now we will divide the whole equation with$2$ on both sides, we get
$\begin {align}
  & \Rightarrow \dfrac{2x}{2}=\dfrac{\dfrac{\ln \left (316 \right)}{\ln \left (7 \right)}+1}{2} \\
 & \Rightarrow x=\dfrac {\dfrac{\ln \left (316 \right)}{\ln \left (7 \right)}+1}{2} \\
\end{align}$
By using scientific calculator we will find the value of $\ln \left (316 \right)$ is equal to the $5.755742214$ and the value of $\ln \left (7 \right)$ is equal to $1.945910149$ and put these values in above equation, we get
$\begin {align}
  & \Rightarrow x=\dfrac{\dfrac{5.755742214}{1.945910149}+1}{2} \\
 & \Rightarrow x=\dfrac{\dfrac{5.755742214+1.945910149}{1.945910149}}{2} \\
 & \Rightarrow x=\dfrac{\dfrac{7.701652363}{1.945910149}}{2} \\
\end{align}$
Now we will divide the $\dfrac{7.701652363}{1.945910149}$, we get
 $\begin {align}
  & \Rightarrow x=\dfrac {3.957866383} {2} \\
 & \Rightarrow x=1.978933191 \\
\end{align}$
Hence by using logarithms rules and some basic rules of algebraic we get the value of $x=1.978933191$

Note:
We should know some basic points on logarithm. Just like those rules which we used in the above questions. Students may go wrong by writing wrong rules and then get confused in solving problems. Always use a scientific calculator while finding these types of values.