Question

Find the value of Log (4+3i)​

Answer 1

Answer:

4 + 3i is 4 – 3i. ( X + Y ) x (X - Y) = X^2 - Y^2

Step-by-step explanation:

Answer 2

Answer: value of $\bold{log(4+3i)=log 5+i tan^{-1}(3/4)}$.

Given: Log (4+3i)​.

To Find: Value of Log (4+3i)​

Step-by-step explanation:

Step 1: Logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if $b^x = n$, in which case one writes $x = log_bn$.

Step 2: Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number $\theta$:

${\displaystyle e^{i \theta }=\cos \theta +i\sin \theta}$

Using Euler's Form,

$cos\theta + i\sin\theta = e^{i\theta}$

We can find the value of angle,

$\theta = tan^{-1} (\frac{3}{4})\\\\sin \theta = 3/5\\\\cos \theta = 4/5$

Step 3: The modulus of (4+3i) is 5.

$(4+3i) = 5(4/5 + 3/5i)\\\\(4+3i) = 5(cos \theta + isin \theta)$

Using Euler's Form,

$cos\theta + isin\theta = e^{i\theta}$

Thus,

$(4+3i)=5e^{i\theta}$

Using log property,

$log[5.e.(i\theta)] = log5 + log[e^{i.\theta}]\\\\=log5+i\theta\\\\=log 5+i tan^{-1}(3/4)$

Hence, correct answer is $\bold{log 5+i tan^{-1}(3/4)}$.

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