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

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

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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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

Step by step solution

Step 1. Given Information

Step 2. Determine True or False.

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Determine whether each of the following statements is true or false:(i) lna+b=lnab(ii) lnax=xlna(iii) lnab=lna-lnb(iv) eab=ea-eb(v) elnx=x(vi)lnex=1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine True or False.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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