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Chapter 3: Problem 62

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. $$4 \ln x+7 \ln y-3 \ln z$$

Short Answer

Expert verified
\(\ln ((x^4 \times y^7) / z^3)\

Step by step solution

01

Apply the power rule

By applying the power rule to each term, we have \(\ln (x^4) + \ln (y^7) - \ln (z^3)\).
02

Apply the product and quotient rules

Then, applying the product rule to the addition terms and the quotient rule to the subtracted term, we get \(\ln (x^4 \times y^7 / z^3)\).
03

Simplify the expression

Considering \(x\), \(y\), and \(z\) as positive real numbers gives us \(\ln ((x^4 \times y^7) / z^3)\).

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Most popular questions from this chapter

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts \((a)-(c) .\) Try generalizing this observation.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

What is an exponential function?

Describe a difference between exponential growth and logistic growth.

Solve and graph the solution set on a number line: \(2 x^{2}+5 x<12 .\)
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