Question

What is the missing exponent in the following equation?
e?/e65 = e/38
(1 point)

Answer 1

To solve the equation \( \frac{e^?}{e^{65}} = \frac{e}{38} \), we can start by simplifying the left side. We can use the property of exponents that states \( \frac{e^a}{e^b} = e^{a-b} \).

Thus, we have:

\[
\frac{e^?}{e^{65}} = e^{? - 65}
\]

Now, we can set the left side equal to the right side:

\[
e^{? - 65} = \frac{e}{38}
\]

The right side can be rewritten using exponential notation where \( \frac{e}{38} = e^{1} \times \frac{1}{38} \). However, it is clearer to leave it as is for our purposes.

To set exponents equal (since they are both bases of \( e \)), we first rewrite the right side as:

\[
e^{1} \div e^{\log(38)} = e^{1 - \log(38)}
\]

Thus we can set the exponents equal to each other:

\[
? - 65 = 1 - \log(38)
\]

Now, we can solve for \( ? \):

\[
? = 1 - \log(38) + 65
\]

Simplifying this gives:

\[
? = 66 - \log(38)
\]

Thus, the missing exponent is \( 66 - \log(38) \). If you're looking for a numerical answer, you'll need to compute \( \log(38) \) (which is approximately \( 1.581 \)).

Therefore, \( ? \approx 66 - 1.581 \approx 64.419 \) (if calculated).

For the final answer in terms of an expression, it would be \( 66 - \log(38) \).