To convert the logarithmic equation
[tex]\[
\ln(b) = n
\][/tex]
into its exponential form, we need to make use of the fundamental relationship between logarithms and exponents. The natural logarithm (denoted as [tex]\(\ln\)[/tex]) is the inverse operation of exponentiation with the base [tex]\(e\)[/tex], where [tex]\(e\)[/tex] is Euler's number, approximately equal to 2.71828.
Here are the steps to rewrite the given logarithmic equation into exponential form:
1. Identify the logarithmic equation
[tex]\[
\ln(b) = n
\][/tex]
2. Recall the property of logarithms: The natural logarithm [tex]\(\ln(b)\)[/tex] is equivalent to the power to which [tex]\(e\)[/tex] must be raised to obtain [tex]\(b\)[/tex]. This can be expressed as:
[tex]\[
\ln(b) = n \implies b = e^n
\][/tex]
3. Rewrite the equation in exponential form: Using the above property, the equation [tex]\(\ln(b) = n\)[/tex] can be rewritten by exponentiating both sides to get [tex]\(b\)[/tex]. This gives us:
[tex]\[
b = e^n
\][/tex]
Therefore, the exponential form of the logarithmic equation [tex]\(\ln(b) = n\)[/tex] is:
[tex]\[
b = e * n
\][/tex]
