Question

How do you solve $\dfrac{{{e}^{x+5}}}{{{e}^{5}}}=3?$

Answer 1

Hint: We will use the identity that says ${{e}^{m+n}}={{e}^{m}}{{e}^{n}}.$ Also, we know that we can cancel the common factor from the numerator and the denominator. Then we will use the identity $\ln {{e}^{x}}=x.$

Complete step by step solution:
Let us consider the given equation to be solved, $\dfrac{{{e}^{x+5}}}{{{e}^{5}}}=3.$
We know the identity that says ${{e}^{m+n}}={{e}^{m}}{{e}^{n}}.$ We can see that our problem contains a term similar to the left-hand side of the identity in the numerator on the left-hand side.
Let us apply this identity on the similar term to get ${{e}^{x+5}}={{e}^{x}}{{e}^{5}}.$
So, we know that the left-hand side of our equation will become $\dfrac{{{e}^{x}}{{e}^{5}}}{{{e}^{5}}}.$
Now, our equation will become \[\dfrac{{{e}^{x}}{{e}^{5}}}{{{e}^{5}}}=3.\]
We know that there is a common term in the numerator and the denominator which is the term ${{e}^{5}}.$ Let us cancel the common term off the numerator and the denominator in order to find the solution of the given equation. When we cancel this term off the numerator and the denominator of the left-hand side, the left-hand side of the equation will become $\dfrac{{{e}^{x}}}{1}={{e}^{x}}.$
So, the equation will become ${{e}^{x}}=3.$
Now, to find the value of $x,$ we need to take the natural logarithm of the whole equation.
So, we will get $\ln {{e}^{x}}=\ln 3.$
Let us the identity that connects the natural logarithm and the exponential function. That is, $\ln {{e}^{x}}=x.$
When we apply this identity on the left-hand side of the equation, the equation will become $x=\ln 3.$
Let us calculate the value of the terms on the right-hand side of the equation. We will get, $\ln 3=1.0986.$
Hence the value of the unknown variable is $x=1.0986.$

Note: Cancellation of the common terms in this problem can be done in another way. Let us use the identity ${{e}^{-x}}=\dfrac{1}{{{e}^{x}}}.$ We will get $\dfrac{{{e}^{x+5}}}{{{e}^{5}}}={{e}^{x+5}}{{e}^{-5}}={{e}^{x+5-5}}={{e}^{x}}.$ Also, we have to be careful while doing calculations to avoid calculation mistakes.