Express two to the power of negative nine over two is equal to one over 16 times root two in its equivalent logarithmic form.
In this question, we’re given an equation and we’re asked to write an equivalent equation in terms of logarithms. To do this, let’s start by recalling the relationship between exponentials and logarithms. We know if 𝑎 to the power of 𝑥 is equal to 𝑦, then we can say that 𝑥 is equal to the logarithm base 𝑎 of 𝑦. This is because the logarithm base 𝑎 function is the inverse function of the exponential function of base 𝑎. It’s also worth noting we assume our value of 𝑎 is positive and not equal to one.
We want to apply this to the equation given. We can see the left-hand side of this equation is already in exponential form. Our value of 𝑎 is two, and 𝑥 is negative nine over two. This then means the right-hand side of this equation is 𝑦: 𝑦 is equal to one over 16 root two. We can then substitute these values into the logarithmic equation, where we’re going to reorder the two sides of the equation. This gives us the log base two of one over 16 root two is equal to negative nine over two, which is our final answer.
Therefore, we were able to rewrite the equation two to the power of negative nine over two is equal to one over 16 root two in a logarithmic form. We showed it was equivalent to the equation the logarithm base two of one over 16 root two is equal to negative nine over two.