What is the missing exponent in the following equation?

Question

e to what power / e to the 65th power equals e to the
38th power

Answer 1

To solve for the missing exponent in the equation \(\frac{e^x}{e^{65}} = e^{38}\), we can use the properties of exponents.

Using the property \(\frac{e^a}{e^b} = e^{a-b}\), we rewrite the left-hand side:

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

Now we can set the exponents equal to each other, since the equation states:

\[ e^{x - 65} = e^{38} \]

This implies:

\[ x - 65 = 38 \]

To solve for \(x\), we add 65 to both sides:

\[ x = 38 + 65 \]

\[ x = 103 \]

Therefore, the missing exponent is \(103\).