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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.2.3b
Chapter 7, Problem 7.2.3b

3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)

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1
Recall the logarithm property that states: \(\ln(a) + \ln(b) = \ln(ab)\). This allows us to combine the sum of logarithms into a single logarithm of the product.
Identify the two expressions inside the logarithms: the first is \(3x^{2} - 9x\) and the second is \(\frac{1}{3}x\).
Multiply the two expressions inside the logarithms: \((3x^{2} - 9x) \times \left(\frac{1}{3}x\right)\).
Simplify the product by distributing and combining like terms: multiply \$3x^{2}$ by \(\frac{1}{3}x\) and $-9x$ by \(\frac{1}{3}x\).
Write the final expression as a single logarithm: \(\ln\left(\text{simplified product}\right)\).

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Key Concepts

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

Properties of Logarithms

Logarithmic properties allow the simplification and combination of logarithmic expressions. Key properties include the product rule (ln a + ln b = ln(ab)), the quotient rule (ln a - ln b = ln(a/b)), and the power rule (ln(a^b) = b ln a). These rules help rewrite multiple logarithmic terms as a single logarithm.
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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function consists of all positive real numbers inside the logarithm. When combining logarithms, it is essential to ensure the resulting argument remains positive, as ln(x) is undefined for x ≤ 0. This affects the validity of the simplified expression.
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Graphs of Logarithmic Functions

Algebraic Simplification

Algebraic simplification involves factoring and reducing expressions inside the logarithms before applying logarithmic properties. For example, factoring 3x² - 9x as 3x(x - 3) can make it easier to combine terms and simplify the overall expression.
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Related Practice
Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)

1
views
Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

1
views
Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

4. b. arcsin(-1/√2)

Textbook Question

75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

g(x) = x(ln x)²