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)
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) $\square$
Solution
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:
\[
\boxed{y \approx 2.1972}
\]
\[
\boxed{x \approx 54.5982}
\]