Problem 78 - Precalculus

Question

Prove the power rule: \(\log _{b} M^{p}=p \log _{b} M .\) Hint: Let \(u=\log _{b} M .\) Write this log in exponential form and find \(\log _{b} M^{p}\)

Step-by-Step Solution

Answer

The power rule for logarithms is proven as \( \log_b M^p = p \log_b M \).

Step 1: Write the Logarithm in Exponential Form

Start by setting the logarithm given in the problem into exponential form. We have \( u = \log_b M \). This can be written in exponential form as \( M = b^u \).

Step 2: Apply the Power Rule for Exponents

Now consider \( M^p \). Since \( M = b^u \), we can substitute it to get \( M^p = (b^u)^p \). Using the power rule for exponents, \((a^m)^n = a^{mn}\), we find \( M^p = b^{up} \).

Step 3: Take the Logarithm of Both Sides

Take the logarithm base \( b \) of \( M^p \). We have \( \log_b M^p = \log_b (b^{up}) \). By the definition of logarithms, \( \log_b (b^x) = x \), thus \( \log_b (b^{up}) = up \).

Step 4: Express the Result in Terms of \( u \)

Substitute back for \( u \), which is \( \log_b M \). Therefore, \( \log_b M^p = up = p\log_b M \).

Step 5: Conclusion

We have shown that \( \log_b M^p = p \log_b M \), confirming the power rule for logarithms.

Key Concepts

Logarithms

Logarithms are the mathematical operation that helps to determine the power to which a base number must be raised to produce a given number. In simpler terms, if you know the result and the base, logarithms allow you to find the exponent. For example, in the equation, \( b^y = x \), \( y \) is the logarithm of \( x \) with base \( b \), written as \( \log_b x = y \).
Logarithms are inverse operations of exponentiation and they have several properties, including:

• Product Rule: \( \log_b(MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b(M/N) = \log_b M - \log_b N \)
• Power Rule: \( \log_b(M^p) = p\log_b M \)

Understanding these rules helps simplify complex logarithmic expressions and solve problems involving powers and exponents more easily.

Exponential Form

Exponential form is a way of expressing numbers by involving exponents. It demonstrates how many times a number, the base, is multiplied by itself. If you have an equation like \( b^n = x \), this is the number \( x \) written in exponential form. In other words, \( x \) is the result of raising \( b \) to the power of \( n \).
Transferring a logarithm into exponential form is essential to solving many math problems. For instance, transforming a logarithmic statement into an exponential equation can make it easier to compute or simplify. Coming from our core example, if \( u = \log_b M \), by transferring this into exponential form, we can write \( M = b^u \). This form helps to visually grasp the relationship between the base, the exponent, and the result, which is pivotal in various applications in both real-world and theoretical problems.

Power Rule for Exponents

The power rule for exponents is a crucial mathematical rule that applies when you raise an exponent to another power. This rule states that when you have a number raised to a power and then raised again to another power, you can multiply the exponents. Mathematically, this is demonstrated as \( (a^m)^n = a^{mn} \).
Applying this rule in the context of logarithms helps simplify expressions like \( (b^u)^p \). By the power rule, \( (b^u)^p = b^{up} \), allowing further manipulation or simplification of an equation.

• It transforms nested exponents into single exponent terms.
• It is useful in algebraic manipulations to solve complex equations.

Mastering this rule is essential for handling equations with multiple exponents and is particularly powerful in proving more complex logarithmic identities or simplifying mathematical expressions in calculus and algebra.