Math / Algebra

QuestionEvaluate the following expressions without using a calculator. (a) $\ln \left(\sqrt[6]{e^{1}}\right)=$ $\square$ (b) $\ln \left(\frac{1}{e^{7}}\right)=$ $\square$ (c) $\ln \left(e^{9}\right)=$ $\square$ (d) $e^{\ln (4)}=$ $\square$ (e) $e^{\ln (\sqrt{5})}=$ $\square$ Question Help: Video 1 Video 2 Message instructor

Studdy Solution

STEP 1

1. The natural logarithm function, $\ln(x)$ , is the inverse of the exponential function $e^x$ .
2. The properties of logarithms and exponents will be used to simplify the expressions.

STEP 2

1. Simplify the expression inside the logarithm for each part.
2. Apply properties of logarithms and exponents to evaluate the expressions.

STEP 3

(a) Simplify $\ln \left(\sqrt[6]{e^{1}}\right)$ :
First, rewrite the expression inside the logarithm using the property of exponents: $\sqrt[6]{e^{1}} = e^{1/6}$ .
$\ln \left(\sqrt[6]{e^{1}}\right) = \ln \left(e^{1/6}\right)$

STEP 4

(a) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{1/6}\right) = \frac{1}{6}$

STEP 5

(b) Simplify $\ln \left(\frac{1}{e^{7}}\right)$ :
Rewrite the expression using the property of exponents: $\frac{1}{e^{7}} = e^{-7}$ .
$\ln \left(\frac{1}{e^{7}}\right) = \ln \left(e^{-7}\right)$

STEP 6

(b) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{-7}\right) = -7$

STEP 7

(c) Simplify $\ln \left(e^{9}\right)$ :
This expression is already in the form $\ln(e^a)$ .

STEP 8

(c) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{9}\right) = 9$

STEP 9

(d) Simplify $e^{\ln (4)}$ :
This expression is already in the form $e^{\ln(a)}$ .

STEP 10

(d) Apply the property of exponents: $e^{\ln(a)} = a$ .
$e^{\ln (4)} = 4$

STEP 11

(e) Simplify $e^{\ln (\sqrt{5})}$ :
This expression is already in the form $e^{\ln(a)}$ .

STEP 12

(e) Apply the property of exponents: $e^{\ln(a)} = a$ .
$e^{\ln (\sqrt{5})} = \sqrt{5}$
The evaluated expressions are: (a) $\frac{1}{6}$ (b) $-7$ (c) $9$ (d) $4$ (e) $\sqrt{5}$

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Studdy Solution

STEP 1

1. The natural logarithm function, $\ln(x)$ , is the inverse of the exponential function $e^x$ .
2. The properties of logarithms and exponents will be used to simplify the expressions.

STEP 2

1. Simplify the expression inside the logarithm for each part.
2. Apply properties of logarithms and exponents to evaluate the expressions.

STEP 3

(a) Simplify $\ln \left(\sqrt[6]{e^{1}}\right)$ :
First, rewrite the expression inside the logarithm using the property of exponents: $\sqrt[6]{e^{1}} = e^{1/6}$ .
$\ln \left(\sqrt[6]{e^{1}}\right) = \ln \left(e^{1/6}\right)$

STEP 4

(a) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{1/6}\right) = \frac{1}{6}$

STEP 5

(b) Simplify $\ln \left(\frac{1}{e^{7}}\right)$ :
Rewrite the expression using the property of exponents: $\frac{1}{e^{7}} = e^{-7}$ .
$\ln \left(\frac{1}{e^{7}}\right) = \ln \left(e^{-7}\right)$

STEP 6

(b) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{-7}\right) = -7$

STEP 7

(c) Simplify $\ln \left(e^{9}\right)$ :
This expression is already in the form $\ln(e^a)$ .

STEP 8

(c) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{9}\right) = 9$

STEP 9

(d) Simplify $e^{\ln (4)}$ :
This expression is already in the form $e^{\ln(a)}$ .

STEP 10

(d) Apply the property of exponents: $e^{\ln(a)} = a$ .
$e^{\ln (4)} = 4$

STEP 11

(e) Simplify $e^{\ln (\sqrt{5})}$ :
This expression is already in the form $e^{\ln(a)}$ .

STEP 12

(e) Apply the property of exponents: $e^{\ln(a)} = a$ .
$e^{\ln (\sqrt{5})} = \sqrt{5}$
The evaluated expressions are: (a) $\frac{1}{6}$ (b) $-7$ (c) $9$ (d) $4$ (e) $\sqrt{5}$

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Studdy Solution

STEP 1

1. The natural logarithm function, ln⁡(x)\ln(x)ln(x), is the inverse of the exponential function exe^xex.2. The properties of logarithms and exponents will be used to simplify the expressions.

STEP 2

1. Simplify the expression inside the logarithm for each part.2. Apply properties of logarithms and exponents to evaluate the expressions.

STEP 3

(a) Simplify ln⁡(e16)\ln \left(\sqrt[6]{e^{1}}\right)ln(6e1​):First, rewrite the expression inside the logarithm using the property of exponents: e16=e1/6\sqrt[6]{e^{1}} = e^{1/6}6e1​=e1/6.ln⁡(e16)=ln⁡(e1/6) \ln \left(\sqrt[6]{e^{1}}\right) = \ln \left(e^{1/6}\right) ln(6e1​)=ln(e1/6)

STEP 4

(a) Apply the property of logarithms: ln⁡(ea)=a\ln(e^a) = aln(ea)=a.ln⁡(e1/6)=16 \ln \left(e^{1/6}\right) = \frac{1}{6} ln(e1/6)=61​

STEP 5

(b) Simplify ln⁡(1e7)\ln \left(\frac{1}{e^{7}}\right)ln(e71​):Rewrite the expression using the property of exponents: 1e7=e−7\frac{1}{e^{7}} = e^{-7}e71​=e−7.ln⁡(1e7)=ln⁡(e−7) \ln \left(\frac{1}{e^{7}}\right) = \ln \left(e^{-7}\right) ln(e71​)=ln(e−7)

STEP 6

(b) Apply the property of logarithms: ln⁡(ea)=a\ln(e^a) = aln(ea)=a.ln⁡(e−7)=−7 \ln \left(e^{-7}\right) = -7 ln(e−7)=−7

STEP 7

(c) Simplify ln⁡(e9)\ln \left(e^{9}\right)ln(e9):This expression is already in the form ln⁡(ea)\ln(e^a)ln(ea).

STEP 8

(c) Apply the property of logarithms: ln⁡(ea)=a\ln(e^a) = aln(ea)=a.ln⁡(e9)=9 \ln \left(e^{9}\right) = 9 ln(e9)=9

STEP 9

(d) Simplify eln⁡(4)e^{\ln (4)}eln(4):This expression is already in the form eln⁡(a)e^{\ln(a)}eln(a).

STEP 10

(d) Apply the property of exponents: eln⁡(a)=ae^{\ln(a)} = aeln(a)=a.eln⁡(4)=4 e^{\ln (4)} = 4 eln(4)=4

STEP 11

(e) Simplify eln⁡(5)e^{\ln (\sqrt{5})}eln(5​):This expression is already in the form eln⁡(a)e^{\ln(a)}eln(a).

STEP 12

(e) Apply the property of exponents: eln⁡(a)=ae^{\ln(a)} = aeln(a)=a.eln⁡(5)=5 e^{\ln (\sqrt{5})} = \sqrt{5} eln(5​)=5​ The evaluated expressions are: (a) 16\frac{1}{6}61​ (b) −7-7−7 (c) 999 (d) 444 (e) 5\sqrt{5}5​

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1. The natural logarithm function, $\ln(x)$ , is the inverse of the exponential function $e^x$ .
2. The properties of logarithms and exponents will be used to simplify the expressions.

1. Simplify the expression inside the logarithm for each part.
2. Apply properties of logarithms and exponents to evaluate the expressions.

(a) Simplify $\ln \left(\sqrt[6]{e^{1}}\right)$ :
First, rewrite the expression inside the logarithm using the property of exponents: $\sqrt[6]{e^{1}} = e^{1/6}$ .
$\ln \left(\sqrt[6]{e^{1}}\right) = \ln \left(e^{1/6}\right)$

(a) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{1/6}\right) = \frac{1}{6}$

(b) Simplify $\ln \left(\frac{1}{e^{7}}\right)$ :
Rewrite the expression using the property of exponents: $\frac{1}{e^{7}} = e^{-7}$ .
$\ln \left(\frac{1}{e^{7}}\right) = \ln \left(e^{-7}\right)$

(b) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{-7}\right) = -7$

(c) Simplify $\ln \left(e^{9}\right)$ :
This expression is already in the form $\ln(e^a)$ .

(c) Apply the property of logarithms: $\ln(e^a) = a$ .
$\ln \left(e^{9}\right) = 9$

(d) Simplify $e^{\ln (4)}$ :
This expression is already in the form $e^{\ln(a)}$ .

(d) Apply the property of exponents: $e^{\ln(a)} = a$ .
$e^{\ln (4)} = 4$

(e) Simplify $e^{\ln (\sqrt{5})}$ :
This expression is already in the form $e^{\ln(a)}$ .

(e) Apply the property of exponents: $e^{\ln(a)} = a$ .
$e^{\ln (\sqrt{5})} = \sqrt{5}$
The evaluated expressions are: (a) $\frac{1}{6}$ (b) $-7$ (c) $9$ (d) $4$ (e) $\sqrt{5}$