Skip to main content
Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 21
Chapter 5, Problem 21

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb (x2 y)

Verified step by step guidance
1
Identify the logarithmic expression to expand: \(\log_{b} (x^{2} y)\).
Use the product property of logarithms, which states that \(\log_{b} (MN) = \log_{b} M + \log_{b} N\), to separate the logarithm of a product into a sum: \(\log_{b} (x^{2} y) = \log_{b} (x^{2}) + \log_{b} (y)\).
Apply the power property of logarithms, which states that \(\log_{b} (M^{k}) = k \log_{b} (M)\), to the term \(\log_{b} (x^{2})\): this becomes \(2 \log_{b} (x)\).
Rewrite the expanded expression combining the results: \(2 \log_{b} (x) + \log_{b} (y)\).
Check if any further simplification or evaluation is possible based on given values or context; if not, the expression is fully expanded.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, log_b(x^2 y) can be expanded using the product rule as log_b(x^2) + log_b(y).
Recommended video:
5:36
Change of Base Property

Product Rule of Logarithms

The product rule states that the logarithm of a product is the sum of the logarithms: log_b(MN) = log_b(M) + log_b(N). This rule helps break down complex expressions into simpler parts, making them easier to evaluate or manipulate.
Recommended video:
3:49
Product, Quotient, and Power Rules of Logs

Power Rule of Logarithms

The power rule states that the logarithm of a power can be expressed as the exponent times the logarithm of the base: log_b(M^k) = k * log_b(M). This is useful for moving exponents in or out of the logarithm to simplify expressions.
Recommended video:
Guided course
5:50
Power Rules
Related Practice
Textbook Question

The graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \(\f\)(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

1
views
Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log16 4

Textbook Question

Evaluate each expression without using a calculator. log4 16

Textbook Question

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. e(x+1)=1/e

4
views
Textbook Question

Write each equation in its equivalent logarithmic form. 7y = 200

Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log5 (1/5)

3
views