To compute the common logarithm of a complex number, we first need to convert it into polar form.
Start by expressing the complex number in the standard form ( a + bi ), where ( a ) and ( b ) are real numbers and ( i ) is the imaginary unit. Next, we calculate the modulus ( r ) and the argument ( \theta ) of the complex number using the following formulas:
r=a2+b2 θ=tan−1(ab)After determining ( r ) and ( \theta ), we can represent the complex number in polar form as:
r(cosθ+isinθ)Finally, we can find the common logarithm of the complex number using the formula:
log(z)=log(r)+iθExample:
Let’s find the common logarithm of the complex number ( 2 + 3i ):
Thus, the complex number in polar form is:
13(cos(56.31∘)+isin(56.31∘))Using the logarithm formula, we can compute the common logarithm:
log(2+3i)=log(13)+i⋅56.31∘To express ( \log(\sqrt{13}) ):
log(13)=21log(13)≈0.1139Therefore, we obtain:
log(2+3i)≈0.1139+i⋅56.31∘To express the argument in terms of natural logarithm:
log(2+3i)=0.1139+i⋅log(10)56.31∘≈0.1139+i⋅0.8686Thus, the common logarithm of ( 2 + 3i ) is approximately:
0.1139+i⋅0.8686 100% |  Global |  97% | |
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Professional Tutors |
| All of our elite tutors are full-time professionals, with at least five years of tuition experience and over 5000 accrued teaching hours in their subject. |
Global |
International Tuition |
| Based in Cambridge, with operations spanning the globe, we can provide our services to support your family anywhere. |
97% |
Independent School Entrance Success |
| Our families consistently gain offers from at least one of their target schools, including Eton, Harrow, Wellington and Wycombe Abbey. |
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