Question

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express the equation in logarithmic form. use ln rather than log.
(a) ( e^{x}=3 ) is equivalent to the logarithmic equation:
( operatorname{in}(3)=x )
(b) ( e^{6}=x ) is equivalent to the logarithmic equation:
( operatorname{in}(x)=6 )

Explanation:

Step1: Recall the relationship between exponential and logarithmic forms

The general relationship is \(a^b = c\) is equivalent to \(\log_a c = b\). When \(a = e\), we use \(\ln\) (natural logarithm), so \(e^b = c\) is equivalent to \(\ln(c)=b\).

Step2: Apply the relationship to part (a)

For \(e^x = 3\), comparing with \(e^b = c\) (here \(b = x\) and \(c = 3\)), using the formula \(\ln(c)=b\), we get \(\ln(3)=x\).

Step3: Apply the relationship to part (b)

For \(e^6 = x\), comparing with \(e^b = c\) (here \(b = 6\) and \(c = x\)), using the formula \(\ln(c)=b\), we get \(\ln(x)=6\).

Answer:

(a) \(\ln(3)=x\)
(b) \(\ln(x)=6\)