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.