Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 37

Chapter 5, Problem 37

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\ln$ $\left$( $\frac{x^3 \sqrt{x^2 + 1}$}{(x + 1)^4} $\right$)$

Verified step by step guidance

Start by writing the given logarithmic expression clearly: $\ln\left( \frac{x^{3} \sqrt{x^{2} + 1}}{(x + 1)^{4}} \right)$.

Use the logarithm property for quotients: $\ln\left( \frac{A}{B} \right) = \ln(A) - \ln(B)$, to separate the expression into two logarithms: $\ln\left(x^{3} \sqrt{x^{2} + 1}\right) - \ln\left((x + 1)^{4}\right)$.

Next, apply the logarithm property for products: $\ln(AB) = \ln(A) + \ln(B)$, to expand $\ln\left(x^{3} \sqrt{x^{2} + 1}\right)$ into $\ln(x^{3}) + \ln\left(\sqrt{x^{2} + 1}\right)$.

Rewrite the square root as an exponent: $\sqrt{x^{2} + 1} = (x^{2} + 1)^{1/2}$, and use the power rule for logarithms: $\ln\left(A^{r}\right) = r \ln(A)$, to express $\ln\left(\sqrt{x^{2} + 1}\right)$ as $\frac{1}{2} \ln(x^{2} + 1)$.

Similarly, apply the power rule to $\ln(x^{3})$ and $\ln\left((x + 1)^{4}\right)$ to get $3 \ln(x)$ and $4 \ln(x + 1)$ respectively. Combine all parts to write the fully expanded expression.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

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 the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), and ln(a^n) = n ln(a). These rules help break down complex expressions into simpler sums and differences of logarithms.

Simplifying Radicals and Exponents

Simplifying expressions involving radicals and exponents is essential before applying logarithmic properties. For instance, the square root √(x² + 1) can be left as is, but recognizing powers like x³ or (x + 1)^4 helps in applying the power rule of logarithms effectively.

Domain Considerations for Logarithmic Functions

The domain of a logarithmic function requires the argument to be positive. When expanding ln[(x³(√(x² + 1)))/(x + 1)⁴], it is important to consider values of x that keep the entire expression inside the logarithm positive to ensure the expression is defined.

Recommended video:

5:26

Graphs of Logarithmic Functions

Related Practice

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^(x+2)=410

views

Textbook Question

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e^x-1+2

Textbook Question

Evaluate each expression without using a calculator. log₄ 1

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

views

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(x3x2+1(x+1)4)\(\ln\) \(\left\)( \(\frac{x^3 \sqrt{x^2 + 1}\)}{(x + 1)^4} \(\right\))ln((x+1)4x3x2+1)

Verified video answer for a similar problem:

Key Concepts

Properties of Logarithms

Simplifying Radicals and Exponents

Domain Considerations for Logarithmic Functions

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\(\ln\) \(\left\)( \(\frac{x^3 \sqrt{x^2 + 1}\)}{(x + 1)^4} \(\right\))$