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Let's go through the math problems one by one:

### 3. Convert each logarithmic equation into exponential form.

**a. \( \log 10,000 = 4 \)**

To convert this logarithmic equation into exponential form, recall that \( \log_b(a) = c \) means \( b^c = a \).

Here, the base \( b \) is 10 (since it's a common logarithm), \( a = 10,000 \), and \( c = 4 \).

\[
10^4 = 10,000
\]

**b. \( \ln 3 = y \)**

The natural logarithm \( \ln \) has a base of \( e \), so we use \( e \) as the base.

\[
e^y = 3
\]

**c. \( \log_2(x) = -1 \)**

Here, the base \( b \) is 2, \( a = x \), and \( c = -1 \).

\[
2^{-1} = x \implies \frac{1}{2} = x
\]

### 4. Convert each exponential equation into logarithmic form.

**a. \( 3^{-2} = \frac{1}{9} \)**

Using the relationship \( b^c = a \Rightarrow \log_b(a) = c \):

\[
\log_3\left(\frac{1}{9}\right) = -2
\]

**b. \( e^3 = 20.0855 \)**

Here, \( b = e \), \( c = 3 \), and \( a = 20.0855 \):

\[
\ln(20.0855) = 3
\]

### 5. Determine the inverse of the following function.

**\( y = -\log(x - 1) + 2 \)**

To find the inverse of a function, follow these steps:

1. Replace \( y \) with \( x \) and solve for \( y \):

\[
x = -\log(y - 1) + 2
\]

2. Isolate the logarithm term:

\[
x - 2 = -\log(y - 1)
\]

3. Multiply both sides by -1:

\[
-(x - 2) = \log(y - 1) \implies \log(y - 1) = 2 - x
\]

4. Convert to exponential form:

\[
10^{2-x} = y - 1
\]

5. Solve for \( y \):

\[
y = 10^{2-x} + 1
\]

So, the inverse function is:

\[
f^{-1}(x) = 10^{2-x} + 1
\]

Would you like more details on any of these solutions, or do you have any further questions?

### Related Questions

1. What is the exponential form of \( \log_5(25) = 2 \)?
2. How would you solve \( \log_4(x) = 3 \) for \( x \)?
3. What is the logarithmic form of \( 2^5 = 32 \)?
4. How do you find the inverse of \( y = \ln(x) + 3 \)?
5. If \( \log_b(x) = 5 \), how do you express \( x \) in exponential form?

### Tip

When converting between logarithmic and exponential forms, remember that \( \log_b(a) = c \) is equivalent to \( b^c = a \). This simple rule is key to understanding and solving many logarithmic and exponential problems.

Convert each logarithmic equation into exponential form. Convert each exponential equation into logarithmic form. Determine the inverse of the function y = -log(x - 1) + 2.

Learn how to convert logarithmic equations into exponential form and vice versa. Understand the process of finding the inverse of a function like y = -log(x - 1) + 2. This lesson covers logarithmic to exponential transformations, inverse function formulas, and related mathematical concepts suitable for Grades 10-12.

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