Question $z = \ln(-10)$ Express in the form of $z = \ln a + bj$ .

Studdy Solution

STEP 1

1. The natural logarithm of a negative number involves complex numbers.
2. The complex logarithm can be expressed in the form $z = \ln |a| + i \arg(a)$ .

STEP 2

1. Express the negative number in polar form.
2. Calculate the magnitude and argument.
3. Express the logarithm in the form $z = \ln a + bj$ .

STEP 3

Express $-10$ in polar form. A complex number can be expressed as $re^{i\theta}$ , where $r$ is the magnitude and $\theta$ is the argument. For $-10$ , the magnitude $r$ is:
$r = |-10| = 10$
The argument $\theta$ is the angle with the positive real axis. Since $-10$ is on the negative real axis, the argument is:
$\theta = \pi$

STEP 4

Calculate the magnitude and argument. We already found:
$r = 10$ $\theta = \pi$

STEP 5

Express the logarithm in the form $z = \ln a + bj$ . Using the formula for the logarithm of a complex number:
$z = \ln |a| + i \arg(a)$
Substitute the values:
$z = \ln(10) + i\pi$
Thus, the expression for $z$ is:
$z = \ln(10) + \pi j$
The expression for $z$ is:
$\boxed{\ln(10) + \pi j}$

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