Let's solve the problem shown in the image.

The given equation is:

$10^a = r$

We are asked to write the equivalent logarithmic equation.

Exponential to Logarithmic Form:

In general, the exponential equation $b^x = y$ is equivalent to the logarithmic form:

$\log_b(y) = x$

Applying it to the problem:

Here, $b = 10$, $x = a$, and $y = r$. So the equivalent logarithmic equation is:

$\log_{10}(r) = a$

Since $\log_{10}$ is the common logarithm, we can also write:

$\log(r) = a$

This is the final answer.

Would you like to dive deeper into how logarithmic transformations work?

Here are 5 related questions for further practice:

1. How do you convert $e^x = y$ to logarithmic form?
2. What is the general rule for rewriting logarithmic functions as exponents?
3. How do base-10 logarithms behave differently from natural logarithms?
4. What happens if $b$ in the equation $b^x = y$ is a fraction?
5. How can logarithmic equations help solve exponential equations in real-life contexts?

Tip: Whenever you're working with logarithmic transformations, remember the properties of logs, like $\log(ab) = \log(a) + \log(b)$, to simplify problems further.