Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.45

Chapter 1, Problem 1.45

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb x/y

Verified step by step guidance

Start by recalling the quotient rule for logarithms: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).

Substitute the given values into the equation: \( \log_b x = 0.36 \) and \( \log_b y = 0.56 \).

Apply the quotient rule: \( \log_b \left( \frac{x}{y} \right) = 0.36 - 0.56 \).

Simplify the expression by performing the subtraction.

The result of the subtraction gives you the value of \( \log_b \left( \frac{x}{y} \right) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Properties

Logarithmic properties are rules that govern the manipulation of logarithms. Key properties include the product rule (log_b(mn) = log_b(m) + log_b(n)), the quotient rule (log_b(m/n) = log_b(m) - log_b(n)), and the power rule (log_b(m^k) = k * log_b(m)). Understanding these properties is essential for simplifying logarithmic expressions and solving equations involving logarithms.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms of different bases, enabling easier calculations or comparisons. It helps in evaluating logarithmic expressions when the base is not easily computable.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves substituting known values into logarithmic equations and applying logarithmic properties to simplify. For example, to evaluate log_b(x/y), one would use the quotient rule to express it as log_b(x) - log_b(y). This process requires a solid understanding of the values of the logarithms involved and how to manipulate them effectively.

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