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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 53
Chapter 5, Problem 53

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. lnxe3\(\ln\]\sqrt\)[3]{\(\frac{x}{e}\)}

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Recognize that the expression involves the natural logarithm of a cube root: \(\ln \sqrt[3]{\frac{x}{e}}\).
Rewrite the cube root as an exponent: \(\ln \left( \frac{x}{e} \right)^{\frac{1}{3}}\).
Use the logarithm power rule to bring the exponent in front: \(\frac{1}{3} \ln \left( \frac{x}{e} \right)\).
Apply the logarithm quotient rule to separate the fraction inside the logarithm: \(\frac{1}{3} \left( \ln x - \ln e \right)\).
Recall that \(\ln e = 1\), so simplify the expression to \(\frac{1}{3} (\ln x - 1)\).

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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 by expanding or condensing them. For example, ln(a/b) = ln(a) - ln(b) and ln(a^r) = r ln(a). These properties are essential for simplifying and expanding logarithmic expressions.
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Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e, where e ≈ 2.718. It is the inverse function of the exponential function e^x. Understanding ln is crucial for manipulating expressions involving e and for applying logarithmic properties correctly.
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Radicals and Exponents

Radicals such as cube roots can be expressed as fractional exponents, e.g., ∛x = x^(1/3). Converting radicals to exponents helps apply logarithmic power rules effectively. This conversion simplifies the expansion of logarithmic expressions involving roots.
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Related Practice
Textbook Question

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log₂ (x + 1)

Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is b. compounded quarterly

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Textbook Question

Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = (½)x and g(x) = (½)x-1 + 1

Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 2 logb x + 3 logb y

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Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is a. compounded semiannually

Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log4(x+5)=3