Problem 2 - Algebra 2 and Trigonometry

Question

If \(\log _{b} c=a,\) explain why \(\log _{b} c^{2}=2 a\)

Step-by-Step Solution

Answer

\(\log_b c^2 = 2a\) because \(c^2 = b^{2a}\) and \(\log_b(b^{2a}) = 2a\).

Step 1: Understanding the Given Information

We are given that \(\log_b c = a\). This means that \(b^a = c\). This is the basic definition of logarithms: the logarithm \(\log_b c\) is the exponent to which the base \(b\) must be raised to obtain \(c\).

Step 2: Expressing \(c^2\) in Terms of \(b\) and \(a\)

Since we know \(b^a = c\), if we square both sides, we get \((b^a)^2 = c^2\). Using the power of a power rule, \((b^a)^2 = b^{2a}\). Thus, \(c^2 = b^{2a}\).

Step 3: Finding the Logarithm of \(c^2\)

We need to find \(\log_b(c^2)\). From the expression \(c^2 = b^{2a}\), we take \(\log_b\) of both sides: \(\log_b(c^2) = \log_b(b^{2a})\).

Step 4: Applying Logarithm Properties

Using the property of logarithms that \(\log_b(b^x) = x\), we find \(\log_b(b^{2a}) = 2a\). Therefore, \(\log_b(c^2) = 2a\).

Key Concepts

Logarithms

Logarithms are a mathematical tool used to determine the exponent needed to raise a base number to a particular value. When we write \(\log_b c = a\), it means the base \(b\) raised to the power of \(a\) will give us \(c\). Logarithms help simplify complex calculations, especially when dealing with exponential relationships. Logarithms have several properties that make them versatile:

• The base \(b\) is a positive number not equal to 1.
• Logarithms convert multiplication into addition, division into subtraction, powers into multiples, and roots into fractions.
• Using the basic equation, \(b^a = c\) corresponds to \(\log_b c = a\).
• There is a special property: \(\log_b(b^x) = x\).

By understanding these properties, we can perform transformations on expressions that involve exponents and logarithms, which is essential for solving equations like \(\log_b c^2 = 2a\).

Exponents

Exponents represent how many times a number, known as the base, is multiplied by itself. Consider \(b^a\), which signifies \(b\) multiplied by itself \(a\) times. Understanding exponents is crucial when dealing with logarithms, since logarithms are the inverse operations of exponents.

• Exponents allow us to express large numbers in a simplified form, such as \(10^3\) for 1000.
• An exponent of zero results in 1: \(b^0 = 1\) for any \(b eq 0\).
• Negative exponents represent division: \(b^{-a} = \frac{1}{b^a}\).
• When multiplying numbers with the same base, add their exponents: \(b^m \times b^n = b^{m+n}\).

To link back to our example, knowing that \(b^a = c\) allows us to infer that raising \(c\) to a power involves the properties of exponents. This understanding leads us smoothly into working out \(c^2 = b^{2a}\).

Power of a Power Rule

The Power of a Power Rule in exponents is used when you're raising an exponent to another power. According to this rule, \((b^m)^n = b^{m \cdot n}\). This simplifies expressions notably and helps in evaluating complex exponential and logarithmic expressions.

• This rule is key when you have nested powers, for instance, \((b^a)^2\) becomes \(b^{2a}\).
• A practical application of this rule is visible when expanding terms like \(c^2 = (b^a)^2\).

In our exercise, applying this rule allowed us to convert \((b^a)^2\) into \(b^{2a}\). This transformation is what makes it possible to then use logarithmic properties to determine that \(\log_b(c^2) = 2a\). The power of a power rule bridges the link between exponents and logarithms by enabling clearer manipulation of these mathematical expressions.