Problem 122 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.

Step-by-Step Solution

Answer

The statement makes sense. Logarithms are a reverse way of expressing exponents and share similar properties such as the product, quotient, and power rules.

Step 1: Understanding the relationship between logarithms and exponents

Logarithms and exponents are closely related. If \(a^b = c\), then the base \(a\) logarithm of \(c\) is \(b\), which can be represented as \( log_{a}c = b \). This relationship shows that logarithms are essentially exponents.

Step 2: Explain the Product Rule for Logarithms and its analogy to Exponents

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of its factors. \( log_b(xy) = log_bx + log_by \). This is analogous to the rule for exponents where the product of bases to exponents is equal to the sum of the exponents (i.e., \( a^m * a^n = a^{(m+n)}\)).

Step 3: Explain the Quotient Rule for Logarithms and its analogy to Exponents

The quotient rule of logarithms states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator: \( log_b(x/y) = log_bx - log_by \). This is analogous to the rule for exponents where the quotient of the bases to exponents is equal to the subtraction of the exponents (i.e., \( a^m / a^n = a^{(m-n)}\)).

Step 4: Explain Power Rule for Logarithms and its analogy to Exponents

The power rule of logarithms states that the logarithm of a base to exponent is equal to the product of the exponent and the logarithm of the base: \( log_b(x^n) = n * log_bx \). This is analogous to the rule for exponents where the power to an exponent can be multiplied (i.e., \( (a^m)^n = a^{m*n} \)).