Question

How do you rewrite ${{y}^{x}}=\dfrac{218}{157}$ in logarithm form?

Answer 1

Hint: We have been given an exponential equation in the above question. To write it in the logarithm form, we need to take the logarithm on both the sides of the given equation. Since the base of the exponential on the LHS of the given equation is equal to y, we have to keep it as the base of the logarithm also. After taking the logarithm, we need to use the logarithm properties given by $\log \left( {{a}^{m}} \right)=m\log a$, ${{\log }_{a}}\left( a \right)=1$, and $\log \left( \dfrac{A}{B} \right)=\log A-\log B$ in order to simplify the logarithm equation obtained.

Complete step-by-step answer:
The exponential equation in the above question is given as
$\Rightarrow {{y}^{x}}=\dfrac{218}{157}$
According to the question, we need to write the above equation in the form of the logarithm function. For this, we have to take the logarithm on the both sides of the above equation. In the above equation, we can see that the LHS is equal to the base of y raised to the power of x. Therefore, the base of the logarithm must be equal to y. So we take the logarithm to the base of y on both the sides of the above equation to get
$\Rightarrow {{\log }_{y}}\left( {{y}^{x}} \right)={{\log }_{y}}\left( \dfrac{218}{157} \right)$
Now, using the property of the logarithm function given by $\log \left( {{a}^{m}} \right)=m\log a$, we can simplify the LHS of the above equation as
$\Rightarrow x{{\log }_{y}}\left( y \right)={{\log }_{y}}\left( \dfrac{218}{157} \right)$
Now, we also know the logarithm property given by ${{\log }_{a}}\left( a \right)=1$. Therefore, the LHS of the above equation is simplified to
$\begin{align}
  & \Rightarrow x\left( 1 \right)={{\log }_{y}}\left( \dfrac{218}{157} \right) \\
 & \Rightarrow x={{\log }_{y}}\left( \dfrac{218}{157} \right) \\
\end{align}$
Now, using the logarithm property given by $\log \left( \dfrac{A}{B} \right)=\log A-\log B$ we can simplify the RHS of the above equation as
$\Rightarrow x={{\log }_{y}}\left( 218 \right)-{{\log }_{y}}\left( 157 \right)$
Hence, we have finally written the given equation in the logarithm form.

Note: For solving the equations of these types, where the variable is an exponent to a number, we need to consider the logarithm on it. So we need to remember all the important properties of the logarithm function in order to simplify it.