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The Road to Reality Study Notes: Difference between revisions
→5.2 The idea of the complex logarithm
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This fact is fundamental in the use of logarithms. To show this, we start with the expression $$b^{m+n} = b^m \times b^n$$. The represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation], which is easy to grasp for $$m$$ and $$n$$ being positive integers, as each side just represents $$m+n$$ instances of the number $$b$$, all multiplied together. If $$b$$ is positive, this law is then showed to hold for exponents that are positive integers, values of 0, and fractions. If $$b$$ is negative, we require further expansion into the complex plane. | This fact is fundamental in the use of logarithms. To show this, we start with the expression $$b^{m+n} = b^m \times b^n$$. The represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation], which is easy to grasp for $$m$$ and $$n$$ being positive integers, as each side just represents $$m+n$$ instances of the number $$b$$, all multiplied together. If $$b$$ is positive, this law is then showed to hold for exponents that are positive integers, values of 0, and fractions. If $$b$$ is negative, we require further expansion into the complex plane. | ||
If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function $$f(z) = b^z$$ such that <math>z=log_bw</math> for $$w=b^z$$ then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>. This would then convert multiplication into addition and allow for exponentiation in the complex plane. | We would need a definition of $$b^p$$ for all complex numbers $$p,q,b$$ such that $$b^{p+q} = b^p \times b^q$$. If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function $$f(z) = b^z$$ such that <math>z=log_bw</math> for $$w=b^z$$ then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>. This would then convert multiplication into addition and allow for exponentiation in the complex plane. | ||
=== 5.3 Multiple valuedness, natural logarithms === | === 5.3 Multiple valuedness, natural logarithms === | ||