Problem 107 - College Algebra Essentials

Question

Describe the power rule for logarithms and give an example.

Step-by-Step Solution

Answer

The power rule for logarithms states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number itself, expressed as \(log_a(M^n) = n \cdot log_a(M)\). For instance, an example of this power rule is \(log_2(8^3) = 3 \cdot log_2(8)\), which equals 9.

Step 1: Definition of Power Rule

The power rule for logarithms, often referred to as the 'third property of logarithms', states that for any positive number \(a\) (where \(a ≠ 1\)), and any real numbers \(M\) and \(n\), the logarithm of a number \(M\) raised to the power \(n\) can be expressed as the product of \(n\) and the logarithm of \(M\). In other words, \(log_a(M^n) = n \cdot log_a(M)\).

Step 2: Application of the Power Rule

Let's put the power rule into practice with a simple example. Suppose we have \(log_2(8^3)\). According to the power rule discussed earlier, this can be rewritten as \(3 \cdot log_2(8)\), since \(n\) is 3 and \(M\) is 8 in this case.

Step 3: Calculation

Now, to calculate this, we know that \(log_2(8) = 3\), because \(2^3 = 8\). Therefore, \(3 \cdot log_2(8)\) equals \(3 \cdot 3 = 9\).