Convert the following exponential expression to its equivalent logarithmic form.
$e^9=x+3$

• A. $\log\left(x+3\right)=9$
• B. $\ln\left(x+3\right)=9$
• C. $\ln9=x+3$
• D. $\log_9x=e^3$

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the relationship between exponential and logarithmic forms. The general rule is that if a^b = c, then log_a(c) = b. For natural logarithms, the base is e, so if e^b = c, then ln(c) = b.
2. Step 2: Identify the given exponential expression. In this case, the expression is e^9 = x + 3.
3. Step 3: Apply the natural logarithm (ln) to both sides of the equation to convert the exponential form to logarithmic form. Using the rule ln(e^b) = b, we get ln(x + 3) = 9.
4. Step 4: Verify the equivalence of the logarithmic form ln(x + 3) = 9 with the original exponential expression e^9 = x + 3. This confirms the conversion is correct.
5. Step 5: Note that the other expressions provided in the problem, such as ln(9) = x + 3 and log_9(x) = e^3, are unrelated to the given exponential expression and do not represent equivalent forms of e^9 = x + 3.