What does a logarithm ask me to find?

It asks for an exponent. In log₂(8), the base is 2 and the number inside is 8: you’re finding the power you put on 2 to get 8. Since 2³ = 8, log₂(8) = 3.

The general translation is log_b(x) = y exactly when bʸ = x. In words: “log base b of x equals y” means “b to the power y equals x.” The logarithm’s answer is the exponent, not the base or the number inside. [S1]

Use this translation pattern. Each line shows logarithmic form, exponential form, and how to read it:

• log₂(8) = 3 ↔ 2³ = 8. Read: 2 to what power makes 8? Three.
• log₃(81) = 4 ↔ 3⁴ = 81. Read: 3 to what power makes 81? Four.
• 5² = 25 ↔ log₅(25) = 2. The exponent on 5 that makes 25 is two.
• 10⁰ = 1 ↔ log₁₀(1) = 0. The exponent on 10 that makes 1 is zero.

The general correspondence is log_b(x) = y ↔ bʸ = x. Keep the base b in the same position; the logarithm’s answer y becomes the exponent, and x becomes the result. These examples apply the standard equivalence. [S1]

Start by translating, then ask whether you already know a power that fits.

Example 1, logarithm to exponential form: solve log₃(x) = 4. Translate it as 3⁴ = x. Since 3⁴ = 81, x = 81. To check, put 81 inside the logarithm: log₃(81) = 4 because 3⁴ = 81.

Example 2, exponential to logarithmic form: rewrite 2⁵ = 32 as log₂(32) = 5. The base remains 2, the result 32 goes inside the logarithm, and the exponent 5 becomes the value of the logarithm. [S1]

A handy fill-in template is:

log_[base]([result]) = [exponent] ↔ [base]^[exponent] = [result]

For instance, with base 4, exponent 3, and result 64: log₄(64) = 3 ↔ 4³ = 64. If you don’t know the matching power, the translation still gives you the right equation to solve; the examples here use exact powers.

Check the roles before calculating: the base is the small number after “log,” the result is inside the parentheses, and the logarithm’s value is the exponent. Translate the statement into exponential form and see whether it matches the original numbers.

For real logarithms, the result inside must be positive, and the base must be positive and not equal to 1. [S1] So log₂(8) is defined, while log₂(0) is not a real logarithm. These rules tell you when the notation is valid; they do not by themselves find an answer when the needed exponent is not an obvious whole number. In that case, you may need a method beyond the exact-power examples here.