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question 4

convert the following logarithmic equation into an equivalent exponential equation:
ln(12) = x² - 3

○ 10^(x² - 3) = 12
○ e^(x² - 3) = 12
○ 10^12 = x² - 3
○ e^12 = x² - 3

Explanation:

Step1: Recall the definition of natural logarithm

The natural logarithm \(\ln(y)\) is defined as \(\log_{e}(y)\), where \(e\) is the base of the natural logarithm. By the definition of logarithms, if \(\log_{b}(a)=c\), then \(b^{c}=a\). For the natural logarithm, \(b = e\), so if \(\ln(a)=c\), then \(e^{c}=a\).

Step2: Apply the definition to the given equation

We are given the equation \(\ln(12)=x^{2}-3\). Using the definition of the natural logarithm (where the base \(b = e\), the argument \(a = 12\), and the value \(c=x^{2}-3\)), we can rewrite the logarithmic equation as an exponential equation. Substituting into the formula \(b^{c}=a\) (with \(b = e\), \(c=x^{2}-3\), and \(a = 12\)), we get \(e^{x^{2}-3}=12\).

Answer:

\(e^{x^{2}-3}=12\) (corresponding to the option \(e^{x^{2}-3}=12\))