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Chapter 4: Q. 3 (page 398)

Determine whether each of the following statements is true or false:
(i) $\ln (a + b) = \ln (a b)$
(ii) $\ln (a^{x}) = x \ln (a)$
(iii) $\ln (\frac{a}{b}) = \ln (a) - \ln (b)$
(iv) $e^{\frac{a}{b}} = e^{a} - e^{b}$
(v) $e^{\ln (x)} = x$
(vi) $\ln (e^{x}) = 1$

Short Answer

Expert verified

(i) True

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

Step by step solution

01

Step 1. Given Information

(i) $\ln (a + b) = \ln (a) + \ln (b)$

(ii) $\ln (a^{x}) = x \ln (a)$

(iii) $\ln (\frac{a}{b}) = \ln (a) - \ln (b)$

(iv) $e^{\frac{a}{b}} = e^{a} - e^{b}$

(v) $e^{\ln (x)} = x$

(vi) $\ln (e^{x}) = 1$

02

Step 2. Determine True or False.

(i) $\ln (a + b) = \ln (a) + \ln (b)$

True as it is a logarithmic formula.

(ii) $\ln (a^{x}) = x \ln (a)$

True as it is a logarithmic formula.

(iii) $\ln (\frac{a}{b}) = \ln (a) - \ln (b)$

True as it is a logarithmic formula.

(iv) $e^{\frac{a}{b}} = e^{a} - e^{b}$

False as $e^{\frac{a}{b}} = \frac{e^{a}}{e^{b}}$ .

(v) $e^{\ln (x)} = x$

True.

$y = e^{\ln (x)} \ln (y) = \ln (e^{\ln (x)}) \ln (y) = \ln (x) \ln (e) \ln (y) = \ln (x) . 1 y = x$

(vi) $\ln (e^{x}) = 1$

False.

$\ln (e^{x}) = 1 \ln (e^{x}) = x \ln (e) = x$

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan (x) = \frac{\sin (x)}{\cos (x)}$ ).

$鈭� x^{2} {(x^{3} + 1)}^{5} d x .$

Write out all the integration formulas and rules that we know at this point.

Approximations and limits: Describe in your own words how the slope of a tangent line can be approximated by the slope of a nearby secant line. Then describe how the derivative of a function at a point is defined as a limit of slopes of secant lines. What is the approximation/limit situation described in this section?

Compare the definitions of the definite and indefinite integrals. List at least three things that are different about these mathematical objects.

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan (x) = \frac{\sin (x)}{\cos (x)}$ ).

$鈭� (\tan (x) + x \sec^{2} (x)) d x .$

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