Question

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

Short answer

Question: Prove the logarithmic identity \(\log _{b} x^{z}=z \log _{b} x\). Answer: We proved that by converting x into the form \(b^p\), expressing \(x^z\) in terms of \(b\) and \(p\), and calculating and simplifying \(\log _{b} x^{z}\), we arrived at the desired result: \(\log _{b} x^{z}=z \log _{b} x\).

Step-by-step solution

Convert x to exponential form

In order to work with logarithms, we will first need to convert the variable \(x\) to the form \(x = b^p\). Since we are given that \(x = b^p\), we can solve for \(p\) by taking the logarithm base b of both sides. Doing so, we get \(\log_b{x} = \log_b{b^p}\). Since \(\log_b{b^p} = p\), we have \(p=\log_{b}{x}\).

Express x^z in terms of b and p

Now that we know that \(x = b^{\log_b{x}}\), we can use property E3 for exponents to express \(x^z\) in terms of \(b\) and \(p\). Using the result from step 1, we have \(x^{z} = (b^{\log_b{x}})^{z} = b^{z\log_b{x}}\).

Compute log_b(x^z) and simplify

Now that we have \(x^z = b^{z\log_b{x}}\), we can compute \(\log_b{x^z}\) by taking the logarithm base b of both sides. Doing so, we get \(\log_b{x^z} = \log_b{b^{z\log_b{x}}}\). Since we know that \(\log_b{b^y} = y\), we can rewrite this expression as \(z\log_b{x}\), which completes the proof. Therefore, \(\log _{b} x^{z}=z \log _{b} x\).

Key concepts

Exponential Form

When dealing with logarithms, one helpful step is to express variables or numbers in exponential form. This means rewriting a number as a power of another number, usually referred to as the base. For example, if you have a number \(x\) and express it as \(x = b^p\), this means that \(x\) is written as an exponent of base \(b\) with the power \(p\).

In the given problem, we were asked to set \(x = b^p\). To find \(p\), you take the logarithm of both sides with base \(b\), resulting in \(p = \log_b{x}\). This step simplifies the expression and is fundamental when working with logarithmic equations. Expressing numbers in exponential form allows for simpler manipulation, especially when applying logarithmic properties later.

Understanding the relationship between logarithms and exponents is crucial. Remember that a logarithm answers the question: to what power must the base be raised, to produce a given number? This interchangeability makes converting between logarithmic and exponential form a powerful tool in mathematics.

Logarithmic Properties

Logarithms have several useful properties that make them easier to work with in mathematical equations. These properties can simplify expressions and help solve complex problems. One such property is the multiplication inside a logarithm becomes an addition outside: \(\log_b(mn) = \log_b{m} + \log_b{n}\). Another is the power rule for logarithms: \(\log_b{(m^n)} = n \cdot \log_b{m}\).

The latter property, which is crucial for this problem, allows you to bring down the exponent as a multiplier. In doing so, it simplifies the process of solving logarithmic equations when powers are involved. In our exercise, we utilized this property when we expressed \(x^z = b^{z\log_b{x}}\).

Remembering these properties can greatly reduce the complexity of logarithmic expressions and calculations. They are especially important when proofs involve changing forms and simplifying expressions as in the given problem. By practicing these rules, you'll find it easier to tackle logarithmic equations in the future.

Proof Techniques

Proof techniques are structured methods used to demonstrate that certain mathematical statements are true. The process usually involves breaking down complex statements into smaller, understandable steps. For instance, in our exercise, a proof technique was employed to show \(\log_{b} x^{z}=z \log_{b} x\).

Proof by example or step-by-step proof is one effective method. It includes starting with known equations or identities, applying logical reasoning and properties, like logarithmic properties, to arrive at the desired conclusion. In our case, step-by-step reasoning was used to manipulate the initial expression using exponential form and logarithmic rules until the statement was simplified to prove that both sides of the equation were equal.

Practicing proofs helps in understanding the underlying logic and structure of mathematical equations. It encourages a deeper comprehension beyond just memorizing formulas, urging you to know how and why these formulas work. With regular practice, proof techniques become an essential part of solving mathematical problems, providing clarity and thorough solution pathways.