53 lines
1.8 KiB
Markdown
53 lines
1.8 KiB
Markdown
https://www.youtube.com/watch?v=sULa9Lc4pck
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$$log_a(b) $$
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Pronounced log *base* a, this function is used to figure out what exponent you need to raise $a$ to to get $b$.
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$log_ab = c$ can be rewritten as $a^c = b$.
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$$ 5^{log_5^{(x+2)}}=x+2 $$
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By default, $log$ refers to $log_{10}$. $ln$ is shorthand for $log_e$.
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$$ \sqrt{x} = x^{1/2} $$
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To get the reciprocal of a value, change the sign of the exponent.
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$$ x^{-1} = \frac{1}{x} $$
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## Domain
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There are 3 places you need to worry about domain. The third is specific to logarithms.
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- You can't divide by 0
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- You can't take the square root of a negative without complex numbers
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- You cannot take the $log$ of a zero, or a negative number.
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- There's no way to raise a number to an exponent and have it equal zero, or be a negative number.
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- This can be used to help solve inequalities, because you know *an equation that's wrapped in a logarithm must be $> 0$*.
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### Finding the domain of added logarithms
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$$ log(x+2) + log(2x-3) $$
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With the above example, you can find the domain of each function separately, then find the overlap of valid numbers.
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# Expanding logarithms
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$$ log_b(x*y) = log_b x + log_b y $$
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$$log_b(\frac{x}{y}) = log_b x - log_b y $$
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Example Problem:
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$$ log_5 z = 3 $$
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$$ log_5 y = 2 $$$$log_5(yz) = log_5 y + log_5 z $$$$ 2 + 3 = 5 $$
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Exponents can be moved to the front of a logarithm
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$$ log_3 x^5 = 5*log_3 x $$
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Roots are just the inverse, so:
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$$ log_3 \sqrt x = \frac{1}{2}*log_3x $$
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## Change of base
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$$ log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b} $$
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The above are all equivalent because the ratios are the same
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### The compound interest formula
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$$ A= Pe^{rt} $$
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| Value | Description |
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| ---- | ---- |
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| $A$ | Ending amount |
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| $P$ or $A_0$ | Starting amount |
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| $r$ or $k$ | Rate (a %) |
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| $t$ | The amount of times interest is compounded |
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