Question

question 3 of 10
which logarithmic equation is equivalent to the exponential equation below?
e^a = 60

a. log_60 4 = e
b. ln a = 60
c. log_a 60 = 4
d. ln 60 = a

Explanation:

Step1: Recall the exponential - logarithmic conversion formula

The general formula for converting an exponential equation \(b^y = x\) to a logarithmic equation is \(\log_b x=y\). Also, when the base \(b = e\), we use the natural logarithm \(\ln\) (where \(\ln x=\log_e x\)).

Given the exponential equation \(e^a = 60\), comparing it with \(b^y=x\) (here \(b = e\), \(y=a\), \(x = 60\)).

Step2: Apply the conversion formula

Using the formula \(\log_b x=y\) (or \(\ln x=y\) when \(b = e\)), for \(e^a=60\), we can rewrite it as \(\ln 60=a\) (since \(\ln\) is \(\log_e\), so \(\log_e60=a\) which is the same as \(\ln 60 = a\)).

Let's check the other options:

• Option A: \(\log_{60}4=e\) would correspond to \(60^e = 4\), which is not related to \(e^a = 60\).
• Option B: \(\ln a=60\) would correspond to \(e^{60}=a\), which is not the same as \(e^a = 60\).
• Option C: \(\log_a60 = 4\) would correspond to \(a^4=60\), which is not related to \(e^a = 60\).

Answer:

D. \(\ln 60 = a\)