Precalculus Examples

$- e^{- 0.1 \cdot 4} - e^{- 0.1 \cdot 3} + 5 (\ln (4) - \ln (3))$

Step 1

Simplify each term.

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Step 1.1

Multiply $- 0.1$ by $4$ .

$- e^{- 0.4} - e^{- 0.1 \cdot 3} + 5 (\ln (4) - \ln (3))$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{e^{0.4}} - e^{- 0.1 \cdot 3} + 5 (\ln (4) - \ln (3))$

Step 1.3

Multiply $- 0.1$ by $3$ .

$- \frac{1}{e^{0.4}} - e^{- 0.3} + 5 (\ln (4) - \ln (3))$

Step 1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} + 5 (\ln (4) - \ln (3))$

Step 1.5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} + 5 \ln (\frac{4}{3})$

Step 1.6

Simplify $5 \ln (\frac{4}{3})$ by moving $5$ inside the logarithm.

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} + \ln ({(\frac{4}{3})}^{5})$

Step 1.7

Apply the product rule to $\frac{4}{3}$ .

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} + \ln (\frac{4^{5}}{3^{5}})$

Step 1.8

Raise $4$ to the power of $5$ .

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} + \ln (\frac{1024}{3^{5}})$

Step 1.9

Raise $3$ to the power of $5$ .

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} + \ln (\frac{1024}{243})$

Step 2

To write $- \frac{1}{e^{0.3}}$ as a fraction with a common denominator, multiply by $\frac{e^{0.1}}{e^{0.1}}$ .

$- \frac{1}{e^{0.4}} - \frac{1}{e^{0.3}} \cdot \frac{e^{0.1}}{e^{0.1}} + \ln (\frac{1024}{243})$

Step 3

Write each expression with a common denominator of $e^{0.4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{e^{0.3}}$ by $\frac{e^{0.1}}{e^{0.1}}$ .

$- \frac{1}{e^{0.4}} - \frac{e^{0.1}}{e^{0.3} e^{0.1}} + \ln (\frac{1024}{243})$

Step 3.2

Multiply $e^{0.3}$ by $e^{0.1}$ by adding the exponents.

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Step 3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{e^{0.4}} - \frac{e^{0.1}}{e^{0.3 + 0.1}} + \ln (\frac{1024}{243})$

Step 3.2.2

Add $0.3$ and $0.1$ .

$- \frac{1}{e^{0.4}} - \frac{e^{0.1}}{e^{0.4}} + \ln (\frac{1024}{243})$

Step 4

Combine the numerators over the common denominator.

$\frac{- 1 - e^{0.1}}{e^{0.4}} + \ln (\frac{1024}{243})$

Step 5

To write $\ln (\frac{1024}{243})$ as a fraction with a common denominator, multiply by $\frac{e^{0.4}}{e^{0.4}}$ .

$\frac{- 1 - e^{0.1}}{e^{0.4}} + \frac{\ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$

Step 6

Combine the numerators over the common denominator.

$\frac{- 1 - e^{0.1} + \ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$

Step 7

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) - e^{0.1} + \ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$

Step 8

Factor $- 1$ out of $- e^{0.1}$ .

$\frac{- 1 (1) - (e^{0.1}) + \ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$

Step 9

Factor $- 1$ out of $- 1 (1) - (e^{0.1})$ .

$\frac{- 1 (1 + e^{0.1}) + \ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$

Step 10

Factor $- 1$ out of $\ln (\frac{1024}{243}) e^{0.4}$ .

$\frac{- 1 (1 + e^{0.1}) - (- \ln (\frac{1024}{243}) e^{0.4})}{e^{0.4}}$

Step 11

Factor $- 1$ out of $- 1 (1 + e^{0.1}) - (- \ln (\frac{1024}{243}) e^{0.4})$ .

$\frac{- 1 (1 + e^{0.1} - \ln (\frac{1024}{243}) e^{0.4})}{e^{0.4}}$

Step 12

Simplify the expression.

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Step 12.1

Move the negative in front of the fraction.

$- \frac{1 + e^{0.1} - \ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$

Step 12.2

Reorder factors in $- \frac{1 + e^{0.1} - \ln (\frac{1024}{243}) e^{0.4}}{e^{0.4}}$ .

$- \frac{1 + e^{0.1} - e^{0.4} \ln (\frac{1024}{243})}{e^{0.4}}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{1 + e^{0.1} - e^{0.4} \ln (\frac{1024}{243})}{e^{0.4}}$

Decimal Form:

$0.02727209 \dots$