Question

Use the following steps to prove that \(\log _{b}\left(x^{y}\right)=y \log _{b} x\). a. Let \(x=b^{p}\). Solve this expression for \(p\). b. Use property E3 for exponents to express \(x^{y}\) in terms of \(b\) and \(p\). c. Compute \(\log _{b} x^{y}\) and simplify.

Short answer

Question: Prove the logarithmic identity \(\log_b(x^y) = y \log_b(x)\). Answer: To prove the identity \(\log_b(x^y) = y \log_b(x)\), we first rewrote \(x\) in terms of the base \(b\) and an exponent \(p\) as \(x = b^p\). We then expressed \(x^y\) in terms of \(b\) and \(p\) using the properties of exponents, resulting in \(x^y = b^{py}\). Finally, we computed \(\log_b(x^y)\) and simplified the expression by substituting our expression for \(p\), which resulted in the desired identity: \(\log_b(x^y) = y \log_b(x)\).

Step-by-step solution

Rewrite x in terms of b and p

We are given that \(x = b^p\). To solve this expression for \(p\), we can take the logarithm with base \(b\) of both sides of the equation: \begin{align*} x &= b^p \\ \log_b(x) &= \log_b(b^p) \end{align*} By the definition of logarithms, the right side simplifies to just \(p\), so we have: $$ \log_b(x) = p $$

Express \(x^y\) in terms of b and p

Now that we have an expression for \(p\), we can use it to write \(x^y\) in terms of \(b\) and \(p\): $$ x^y = (b^p)^y $$ Applying the property E3 for exponents (which states that \((a^m)^n = a^{mn}\)), the expression becomes: $$ x^y = b^{py} $$

Compute \(\log_b(x^y)\) and simplify

Now that we have an expression for \(x^y\) in terms of \(b\) and \(p\), we can compute \(\log_b(x^y)\): \begin{align*} \log_b(x^y) &= \log_b(b^{py}) \end{align*} By the definition of logarithms, the right side simplifies as follows: $$ \log_b(x^y) = py $$ Now, we can substitute our expression for \(p\) from Step 1: $$ \log_b(x^y) = (y)(\log_b(x)) $$ This proves the desired identity: $$ \log_b(x^y) = y \log_b(x) $$

Key concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that can be thought of as repeated multiplication. When we say a number, such as \( x \), is raised to the power of \( y \), written as \( x^y \), it means multiplying \( x \), by itself \( y \) times. This concept is essential for understanding how expressions can be simplified using logarithms.

Consider the expression \((b^p)^y\). Using the properties of exponents, specifically the property E3, which states \((a^m)^n = a^{mn}\), we can simplify this to \(b^{py}\). This transformation is crucial in progressing through logarithmic proofs, as it allows us to match expressions in exponential and logarithmic forms to solve equations or identities easily.

In the realm of exponential functions, it's important to recognize how closely they're tied to logarithmic functions. Exponentials and logarithms are inverse operations. This means when one operation is applied to the other, it simply returns the original value, as seen when solving equations in the exercise.

Logarithmic Identities

Logarithmic identities are rules that apply to logarithmic expressions, making them easier to understand and manipulate. One of the foundational identities used in the exercise is that \ \(\log_b(b^x) = x\). This identity clarifies what happens when you take the logarithm of an exponent that has the same base as the logarithm.

Another crucial identity seen in the exercise is \ \(\log_b(x^y) = y \cdot \log_b(x)\). This equality is derived using the property of exponents and is central to many logarithmic transformations. It allows us to pull out the exponent as a coefficient, simplifying expressions and solving equations.

These identities not only simplify complex equations but also provide powerful tools in calculus and algebra, enabling solutions to otherwise complicated problems. Understanding how these identities work allows students to approach problems more strategically and with confidence.

Mathematical Proof

Mathematical proof is a logical argument that establishes the truth of a mathematical statement. Proofs can thoughtfully demonstrate why certain mathematical properties hold true, such as the logarithmic identity \(\log_b(x^y) = y \log_b(x)\).

In constructing a proof, it's crucial to use defined rules and properties systematically. In our example, the steps included rewriting the variable \(x\) in terms of another expression, applying known identities, and simplifying step-by-step until reaching the desired result. This not only helps in confirming the identity but also strengthens the understanding of the relationships between logarithms and exponents.

Proofs often rely on a blend of algebra, deductive reasoning, and known formulas. Properly following these methods requires patience and practice, as each step must logically follow from the previous one. Once mastered, these techniques provide a deep understanding of mathematical concepts and the ability to verify different mathematical scenarios.