vault backup: 2024-08-28 10:39:34

education/math/MATH1050/Exponents/Logarithms.md

@@ -0,0 +1,52 @@
+ https://www.youtube.com/watch?v=sULa9Lc4pck
+
+$$log_a(b) $$
+Pronounced log *base* a, this function is used to figure out what exponent you need to raise $a$ to to get $b$.
+
+$log_ab = c$ can be rewritten as $a^c = b$.
+
+
+$$ 5^{log_5^{(x+2)}}=x+2 $$
+By default, $log$ refers to $log_{10}$. $ln$ is shorthand for $log_e$. 
+
+$$ \sqrt{x} = x^{1/2} $$
+To get the reciprocal of a value, change the sign of the exponent.
+$$ x^{-1} = \frac{1}{x} $$
+## Domain
+There are 3 places you need to worry about domain. The third is specific to logarithms.
+- You can't divide by 0
+- You can't take the square root of a negative without complex numbers
+- You cannot take the $log$ of a zero, or a negative number.
+	- There's no way to raise a number to an exponent and have it equal zero, or be a negative number.
+	- This can be used to help solve inequalities, because you know *an equation that's wrapped in a logarithm must be $> 0$*.
+
+### Finding the domain of added logarithms
+$$ log(x+2) + log(2x-3) $$
+With the above example, you can find the domain of each function separately, then find the overlap of valid numbers.
+
+
+# Expanding logarithms
+$$ log_b(x*y) = log_b x + log_b y $$
+$$log_b(\frac{x}{y}) = log_b x - log_b y $$
+
+Example Problem:
+$$ log_5 z = 3 $$
+$$ log_5 y = 2 $$$$log_5(yz) = log_5 y + log_5 z $$$$ 2 + 3 = 5 $$
+Exponents can be moved to the front of a logarithm
+$$ log_3 x^5 = 5*log_3 x $$
+Roots are just the inverse, so:
+$$ log_3 \sqrt x = \frac{1}{2}*log_3x $$
+## Change of base
+$$ log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b} $$
+The above are all equivalent because the ratios are the same
+
+
+### The compound interest formula
+$$ A= Pe^{rt} $$
+
+| Value | Description |
+| ---- | ---- |
+| $A$ | Ending amount |
+| $P$ or $A_0$ | Starting amount |
+| $r$ or $k$ | Rate (a %) |
+| $t$ | The amount of times interest is compounded |