What is the value of e^{ln(5)}arctan(5e^{-ln(5)}) ?

The value of e^{ln(5)}arctan(5e^{-ln(5)}) is (5pi)/4

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e^{ln(5)}arctan(5e^{-ln(5)})

e^{l n (5)} arctan (5 e^{- l n (5)})

\frac{5 π}{4}

Decimal

3.92699 \dots

Solution steps

e^{l n (5)} arctan (5 e^{- l n (5)})

Simplify:

5 e^{- l n (5)} = 1

5 e^{- l n (5)}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

e^{- l n (5)} = \frac{1}{e ^{l n (5)}}

= 5 \cdot \frac{1}{e ^{l n (5)}}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 5}{e ^{l n (5)}}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{e ^{l n (5)}}

e^{l n (5)} = 5

e^{l n (5)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 5

= \frac{5}{5}

Apply rule

\frac{a}{a} = 1

= 1

e^{l n (5)} arctan (1) = 5 arctan (1)

e^{l n (5)} arctan (1)

e^{l n (5)} = 5

e^{l n (5)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 5

= 5 arctan (1)

= 5 arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

arctan (1)

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4}

= 5 \cdot \frac{π}{4}

Simplify

5 \cdot \frac{π}{4} : \frac{5 π}{4}

5 \cdot \frac{π}{4}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 5}{4}

= \frac{5 π}{4}

What is the value of e^{ln(5)}arctan(5e^{-ln(5)}) ?
The value of e^{ln(5)}arctan(5e^{-ln(5)}) is (5pi)/4