Example 1: evaluating simple logarithms

• Evaluate log₂ 8, log₃ (1/9), log₅ 5 and log₇ 1.
• Solution log₂ 8 = 3 because 2³ = 8. log₃ (1/9) = −2 because 3⁻² = 1/9. log₅ 5 = 1. log₇ 1 = 0.

Example 2: solving a logarithmic equation

Example 3: an expression inside the logarithm

• Solve log₃ (x − 1) = 2.
• Solution x − 1 = 3² = 9, so x = 10. Since x − 1 = 9 > 0, the solution is accepted.

Example 4: the domain of a logarithmic function

• Find the domain of f(x) = log₂ (x² − 4).
• Solution We need x² − 4 > 0, that is (x − 2)(x + 2) > 0. The roots are −2 and 2, and the product is positive outside them. So the domain is ]−∞, −2[ ∪ ]2, ∞[.

Example 5: an equation with a fractional exponent

Example 6: using the basic property

1. Find the value of 5^{log₅ 12}.
2. By the property a^{log_a x} = x, the value is 12.