We are given the logarithmic equation:

$\log_8(x) - 10 = -7$

To solve for $x$, follow these steps:

Step 1: Isolate the logarithmic term.

Add 10 to both sides:

$\log_8(x) = -7 + 10$

$\log_8(x) = 3$

Step 2: Rewrite the logarithmic equation in exponential form.

The logarithmic equation $\log_b(a) = c$ is equivalent to the exponential form $b^c = a$.

Here, $b = 8$, $c = 3$, and $a = x$. So we can write:

$8^3 = x$

Step 3: Simplify the exponential expression.

$8^3 = 512$

So, $x = 512$.

Final Answer:

$x = 512$

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Would you like further clarification or details on any step?

Related questions:

1. How do you solve a logarithmic equation when the base is different from 10 or $e$?
2. What are some properties of logarithms that could help simplify more complex equations?
3. How can we check if a logarithmic solution is valid in the real number domain?
4. How do you solve logarithmic equations with a different base, such as natural logarithms?
5. What is the inverse relationship between logarithmic and exponential functions?

Tip:

Always remember that logarithms and exponents are inverses of each other. This relationship allows you to convert between the two forms, making it easier to solve equations!