Logarithms Review

A logarithm is simply the inverse of an exponential function. It answers the question: "What exponent do I need to raise a specific base to, to get a certain number?"

The Relationship Between Exponents and Logarithms

An exponential equation and a logarithmic equation can express the exact same relationship.

The exponential statement:

$$2^4 = 16$$

is perfectly equivalent to the logarithmic statement:

$$\log_{2}(16) = 4$$

Both statements express the same fact: the base 2 must be raised to the power of 4 to get the number 16. "What power do I need to raise 2 to, to get 16?" 4.

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Example

Let's evaluate the expression:

$$\log_{3}(81) = x$$

This is asking, "To what power ($x$) must we raise the base 3 to get 81?"

We can rewrite this as an exponential equation:

$$3^x = 81$$

By testing powers of 3, we find:

$$3 \times 3 \times 3 \times 3 = 3^4 = 81$$

Therefore, $x=4$.

$$\log_{3}(81) = 4$$