Logarithmic and exponential relationship displayed as y = log_b(x) and b^y = x.

The image shows two equations related to logarithms:

1. $ y = \log_b(x) $
2. $ b^y = x $

These two equations are equivalent and represent the definition of a logarithm. Let's break this down and explain the relationship.

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🔹 Understanding the Logarithmic Definition

The expression:
$$
y = \log_b(x)
$$
means:
> "To what power must the base $ b $ be raised to get $ x $? The answer is $ y $."

This is equivalent to the exponential form:
$$
b^y = x
$$

So, these two forms are just different ways of expressing the same relationship.

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🔹 Example

Let’s say:
$$
y = \log_2(8)
$$

We ask: "2 raised to what power gives 8?"

Since $ 2^3 = 8 $, then:
$$
y = 3
$$

And in exponential form:
$$
2^3 = 8
$$

So:
- $ y = \log_2(8) = 3 $
- $ 2^3 = 8 $

This confirms that:
$$
\log_b(x) = y \quad \text{if and only if} \quad b^y = x
$$

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🔹 Key Points

- $ b $ is the base (must be positive and $ b \ne 1 $).
- $ x $ must be positive (since logs of non-positive numbers are undefined in real numbers).
- $ y $ is the exponent that makes $ b^y = x $.

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✔ Summary

The image illustrates the fundamental definition of logarithms:
> $ y = \log_b(x) $ if and only if $ b^y = x $

This equivalence is crucial for solving logarithmic and exponential equations, changing bases, and understanding inverse functions (since logarithms and exponentials are inverses).

Let me know if you'd like help solving a specific problem using this concept!