Logarithm Rules 2

CONTINUATION OF THE BASIC PROPERTIES OF LOGARITHMS

In the first set of logarithm rules, we learned the basic rules for products, quotients, powers, change of base, and the special cases of the logarithm of unity and the logarithm of the base. We will now deepen our understanding of these rules, especially in the context of composite expressions, negative exponents, roots, the absolute value, and taking the logarithm of both sides of an equation.

LOGARITHMS OF ROOTS

Since a root can be written as a power, we use the power rule:
√x = x^(1/2), so: log_b(√x) = log_b(x^(1/2)) = (1/2) * log_b(x)
For an n-th root: ⁿ√x = x^(1/n), so: log_b(ⁿ√x) = (1/n) * log_b(x)
Example:
log₃(√9) = log₃(9^(1/2)) = (1/2) * log₃(9) = (1/2) * 2 = 1

NEGATIVE EXPONENT IN THE LOGARITHM

If we have a logarithm of a power with a negative exponent, we also apply the power rule:
log_b(x⁻ⁿ) = -n * log_b(x)
Example:
log₁₀(10⁻²) = -2 * log₁₀(10) = -2 * 1 = -2

LOGARITHM OF THE ABSOLUTE VALUE

The logarithm is defined only for positive arguments; therefore, in some contexts, we use the absolute value:
When the general form of the function is expressed, it is often the case that for an expression like log_b(|x|), the definition is valid for negative values of x because we are only taking the logarithm of the positive part (the absolute value).
Example:
log₁₀(-5) → not defined in real numbers
log₁₀(|-5|) = log₁₀(5) → defined

TAKING THE LOGARITHM OF BOTH SIDES OF AN EQUATION

An important technique for solving equations where the unknown is in the exponent is to take the logarithm of both sides:
If a^x = b, then we can apply a logarithm of any convenient base (commonly log base 10, or ln base e) to both sides:
log(a^x) = log(b)
Using the power rule, this becomes:
x * log(a) = log(b)
So: x = log(b) / log(a)
Example:
3^x = 81
Take log base 10 of both sides: log₁₀(3^x) = log₁₀(81)
Apply the power rule: x * log₁₀(3) = log₁₀(81)
Solve for x: x = log₁₀(81) / log₁₀(3) ≈ 1.908 / 0.477 ≈ 4
(Alternatively, notice 81 = 3⁴, so 3^x = 3⁴, thus x = 4 directly by equating exponents if bases are the same).

REPETITIONS AND HIGHLIGHTS

• A root can be written as a power with the exponent 1/n (e.g., ⁿ√x = x^(1/n)).
• The logarithm of a negative number is not defined in real numbers.
• In expressions involving a power inside a logarithm, the exponent is brought down in front of the logarithm as a multiplier.
• Taking the logarithm of both sides of an equation is key to solving exponential equations.

CONCLUSION

The second group of rules for logarithms extends basic knowledge towards solving more complex expressions and equations. By using logarithms of roots, negative exponents, the absolute value, and by taking the logarithm of both sides of equations, we gain advanced tools that are essential for working with exponential expressions and the logarithmic function. These rules are particularly important for analytical problem-solving, data modeling, and preparing for more complex mathematical content.