Questions: Exponential and Logarithmic Functions Converting between natural logarithmic and exponential equations Rewrite each equation as requested. (a) Rewrite as a logarithmic equation. e^y=9 (b) Rewrite as an exponential equation. ln x=4 (a) (b)

Exponential and Logarithmic Functions
Converting between natural logarithmic and exponential equations

Rewrite each equation as requested.
(a) Rewrite as a logarithmic equation.
e^y=9
(b) Rewrite as an exponential equation.
ln x=4
(a) 
(b)
Transcript text: Exponential and Logarithmic Functions Converting between natural logarithmic and exponential equations Rewrite each equation as requested. (a) Rewrite as a logarithmic equation. \[ e^{y}=9 \] (b) Rewrite as an exponential equation. \[ \ln x=4 \] (a) $\square$ (b) $\square$
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Solution

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Solution Steps

Solution Approach

To convert between exponential and logarithmic equations, use the relationships: \( e^y = x \) is equivalent to \( \ln x = y \), and vice versa. For part (a), convert the exponential equation \( e^y = 9 \) to a logarithmic form. For part (b), convert the logarithmic equation \( \ln x = 4 \) to an exponential form.

Step 1: Convert Exponential to Logarithmic Form

Given the equation \( e^y = 9 \), we can rewrite it in logarithmic form. The equivalent logarithmic equation is: \[ y = \ln(9) \approx 2.1972 \]

Step 2: Convert Logarithmic to Exponential Form

For the equation \( \ln x = 4 \), we convert it to exponential form. The equivalent exponential equation is: \[ x = e^4 \approx 54.5982 \]

Final Answer

The answers to the sub-questions are: (a) \( y \approx 2.1972 \)
(b) \( x \approx 54.5982 \)

Thus, the final boxed answers are: \[ \boxed{y \approx 2.1972} \] \[ \boxed{x \approx 54.5982} \]

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