Question

Convert the following into exponential form:
${{\log }_{a}}\dfrac{1}{a}=-1$

Answer 1

Hint: The expression given in the above form is a logarithmic expression and we are asked to convert this expression into an exponential form. We know that from the logarithmic properties that if we have a logarithmic expression in the following way: ${{\log }_{c}}y=b$ then the exponential expression for this logarithmic expression is equal to${{c}^{b}}=y$. So, using this property in the above equation, we will get the exponential form.

Complete step by step solution:
The equation given in the above problem is as follows:
${{\log }_{a}}\dfrac{1}{a}=-1$
As you can see that the above expression is the logarithm form but we want an exponential form so we are going to use the following property of the logarithm to exponential conversion:
If we have given the following logarithmic expression:
${{\log }_{c}}y=b$
Then the exponential form is as follows:
${{c}^{b}}=y$
Similarly, we can write the exponential form for the given logarithmic expression. For that, substituting “c” as $''a''$, “b” as -1 and y as $\dfrac{1}{a}$ in the above equation and we get,
${{a}^{-1}}=\dfrac{1}{a}$
From the above, you can see that we have written the given logarithmic expression into exponential form.

Note: You can check the exponential form which you have written above is correct or not using the following trick:
The logarithm expression given in the above problem is as follows:
${{\log }_{a}}\dfrac{1}{a}=-1$
And the exponential form which we have written in the above solution is equal to:
${{a}^{-1}}=\dfrac{1}{a}$
Now, put argument as ${{a}^{-1}}$ in log and base as $''a''$ and then see what expression you are getting:
${{\log }_{a}}{{a}^{-1}}$
We know that, if we have a negative 1 as the exponent then we can remove the negative exponent by writing the reciprocal of that number so we can write ${{a}^{-1}}$ as $\dfrac{1}{a}$ in the above logarithm expression and we get,
${{\log }_{a}}\dfrac{1}{a}$
As you can see we are getting the same expression which we have started with so the exponential form written in the above solution is correct.