Problem 75 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the definition of a logarithm to prove $$ \log _{b} m^{n}=n \log _{b} m $$

Step-by-Step Solution

Answer

Using the definition of a logarithm, we first write the equation \(\log _{b} m^{n}\) in exponential form as \(b^{\log _{b} m^{n}}=m^{n}\). Applying the Power Rule of Exponents, we get \(\left(b^{\log _{b} m}\right)^{n}=m^{n}\). By converting the exponential expression inside the parenthesis back to a logarithmic expression, we obtain \(m^{n}=m^{n}\), confirming the equality of the left-hand side and right-hand side. Thus, we have proved the logarithmic identity \(\log _{b} m^{n}=n \log _{b} m\).

Step 1: Write the logarithm in exponential form

Using the definition of logarithm, we will write the equation \(\log _{b} m^{n}\) in the exponential form. By definition, we can write the equation as: $$ b^{\log _{b} m^{n}}=m^{n} $$

Step 2: Apply the Power Rule

Now we will use the Power Rule of Exponents. It states that \(b^{xy} = (b^x)^y\). Applying the Power Rule to our exponential equation, we get: $$ \left(b^{\log _{b} m}\right)^{n}=m^{n} $$

Step 3: Use the logarithmic definition again

Now, we will use the definition of a logarithm again to convert the exponential expression inside the parenthesis back to a logarithmic expression. So, using the definition \(\log_{b}{x} = y\), if \(b^{\log _{b}{m}} = m\), we get: $$ m^{n}=m^{n} $$

Step 4: Conclude the Proof

Since the left-hand side and right-hand side of the equation are equal, we can conclude that: $$ \log _{b} m^{n}=n \log _{b} m $$ The logarithmic identity \(\log _{b} m^{n}=n \log _{b} m\) has been proved using the definition of a logarithm and the Power Rule of Exponents.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. In simple terms, it refers to how many times the base number is multiplied by itself. For instance, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. This means \(a\) is multiplied by itself \(n\) times.

• If \(n\) is 2, it is called squaring.
• If \(n\) is 3, it is referred to as cubing.

Exponentiation is a fundamental concept in algebra and is widely used in various fields of mathematics and science.
Understanding exponentiation is crucial when dealing with logarithms because these operations are inverse processes. When you take a logarithm, you are essentially asking, "To what power must the base be raised to produce a certain number?". The correct application of exponentiation, as in many mathematical proofs like the one above, allows for effective manipulation and simplification of expressions.

Power Rule

The Power Rule is a principle that simplifies the manipulation of expressions involving exponents. It states that for any base \(b\) and exponents \(x\) and \(y\), \(b^{xy} = (b^x)^y\). This rule is very helpful when you need to condense multiple exponentiations into a single expression.

• This rule means that raising a power to another power multiplies the exponents.
• The Power Rule applies not just to numbers, but also to expressions involving variables and constants.

In the context of logarithms, the Power Rule is used to establish identities like \(\log_b m^n = n \log_b m\). This identity allows us to efficiently break down complex logarithmic expressions into simpler components.
Understanding and applying the Power Rule in proofs or problem-solving can greatly enhance one's ability to break down and simplify exponential and logarithmic expressions effectively.

Logarithm Definition

The logarithm is a mathematical function that serves as the inverse to exponentiation. The logarithm \(\log_b a\) is the power to which the base \(b\) must be raised to yield the number \(a\). For example, \(\log_2 8 = 3\) because \(2^3 = 8\).
Logarithms have several properties that facilitate their use in algebra and calculus, especially when dealing with exponential growth models or multiplicative patterns in data.

• One fundamental property is that \(\log_b b = 1\) because \(b^1 = b\).
• Another key property is \(\log_b 1 = 0\) because \(b^0 = 1\).

These properties, along with the definition, assist in proving equations involving logarithmic identities, such as \(\log_b m^n = n \log_b m\). In this proof, using the definition of logarithm helps convert between exponential and logarithmic forms.
Understanding the definition and properties of logarithms is key to solving complex algebraic problems and proofs that involve changing expressions between these two forms.