![log_b{162} = x + 4y\\\\log_b324 = 2x+4y\\\\log_b\frac{8}{9} = 3x-2y\\\\\frac{log_b27}{log_b16} = 3y-4x](https://tex.z-dn.net/?f=log_b%7B162%7D%20%3D%20x%20%2B%204y%5C%5C%5C%5Clog_b324%20%3D%202x%2B4y%5C%5C%5C%5Clog_b%5Cfrac%7B8%7D%7B9%7D%20%3D%203x-2y%5C%5C%5C%5C%5Cfrac%7Blog_b27%7D%7Blog_b16%7D%20%3D%203y-4x)
<em><u>Solution:</u></em>
Given that,
![log_b2 = x\\\\log_b3 = y --------(i)](https://tex.z-dn.net/?f=log_b2%20%3D%20x%5C%5C%5C%5Clog_b3%20%3D%20y%20--------%28i%29)
<em><u>Use the following log rules</u></em>
Rule 1: ![log_b(ac) = log_ba + log_bc](https://tex.z-dn.net/?f=log_b%28ac%29%20%3D%20log_ba%20%2B%20log_bc)
Rule 2: ![log_b\frac{a}{c} = log_ba - log_bc](https://tex.z-dn.net/?f=log_b%5Cfrac%7Ba%7D%7Bc%7D%20%3D%20log_ba%20-%20log_bc)
Rule 3: ![log_ba^c = clog_ba](https://tex.z-dn.net/?f=log_ba%5Ec%20%3D%20clog_ba)
![a) log_b{162}](https://tex.z-dn.net/?f=a%29%20log_b%7B162%7D)
Break 162 down to primes:
![162 = 2^1 \times 3^4](https://tex.z-dn.net/?f=162%20%3D%202%5E1%20%5Ctimes%203%5E4)
![log_b{162} =log_b 2^1. 3^4\\\\By\ rule\ 1\\\\ log_b{162} = log_b 2^1 +log_b 3^4\\\\By\ rule\ 3\\\\1log_b2 + 4log_b3\\\\1x+4y\\\\x+4y](https://tex.z-dn.net/?f=log_b%7B162%7D%20%3Dlog_b%202%5E1.%203%5E4%5C%5C%5C%5CBy%5C%20rule%5C%201%5C%5C%5C%5C%20log_b%7B162%7D%20%3D%20log_b%202%5E1%20%2Blog_b%203%5E4%5C%5C%5C%5CBy%5C%20rule%5C%203%5C%5C%5C%5C1log_b2%20%2B%204log_b3%5C%5C%5C%5C1x%2B4y%5C%5C%5C%5Cx%2B4y)
Thus we get,
![log_b162 = x + 4y](https://tex.z-dn.net/?f=log_b162%20%3D%20x%20%2B%204y)
Next
![b) log_b 324](https://tex.z-dn.net/?f=b%29%20log_b%20324)
Break 324 down to primes:
![324 = 2^2 \times 3^4](https://tex.z-dn.net/?f=324%20%3D%202%5E2%20%5Ctimes%203%5E4)
![log_b324 = log_b 2^2.3^4\\\\By\ rule\ 1\\\\log_b324 = log_b2^2 + log_b3^4\\\\By\ rule\ 3\\\\log_b324 = 2log_b2 + 4log_b3\\\\From\ (i)\\\\log_b324 = 2x + 4y](https://tex.z-dn.net/?f=log_b324%20%3D%20log_b%202%5E2.3%5E4%5C%5C%5C%5CBy%5C%20rule%5C%201%5C%5C%5C%5Clog_b324%20%3D%20log_b2%5E2%20%2B%20log_b3%5E4%5C%5C%5C%5CBy%5C%20rule%5C%203%5C%5C%5C%5Clog_b324%20%3D%202log_b2%20%2B%204log_b3%5C%5C%5C%5CFrom%5C%20%28i%29%5C%5C%5C%5Clog_b324%20%3D%202x%20%2B%204y)
Next
![c) log_b\frac{8}{9}](https://tex.z-dn.net/?f=c%29%20log_b%5Cfrac%7B8%7D%7B9%7D)
By rule 2
![log_b\frac{8}{9} = log_b8 - log_b9\\\\log_b\frac{8}{9} = log_b 2^3 - log_b3^2\\\\By\ rule\ 3\\\\log_b\frac{8}{9} = 3 log_b2 - 2log_b3\\\\From\ (i)\\\\log_b\frac{8}{9} = 3x - 2y](https://tex.z-dn.net/?f=log_b%5Cfrac%7B8%7D%7B9%7D%20%3D%20log_b8%20-%20log_b9%5C%5C%5C%5Clog_b%5Cfrac%7B8%7D%7B9%7D%20%3D%20log_b%202%5E3%20-%20log_b3%5E2%5C%5C%5C%5CBy%5C%20rule%5C%203%5C%5C%5C%5Clog_b%5Cfrac%7B8%7D%7B9%7D%20%3D%20%203%20log_b2%20-%202log_b3%5C%5C%5C%5CFrom%5C%20%28i%29%5C%5C%5C%5Clog_b%5Cfrac%7B8%7D%7B9%7D%20%3D%20%203x%20-%202y)
Next
![d) \frac{log_b27}{log_b16}](https://tex.z-dn.net/?f=d%29%20%5Cfrac%7Blog_b27%7D%7Blog_b16%7D)
By rule 2
![\frac{log_b27}{log_b16} = log_b27 - log_b16\\\\ \frac{log_b27}{log_b16} = log_b3^3 - log_b2^4\\\\By\ rule\ 2\\\\ \frac{log_b27}{log_b16} = 3log_b3 - 4log_b2 \\\\From\ (i)\\\\\frac{log_b27}{log_b16} = 3y - 4x](https://tex.z-dn.net/?f=%5Cfrac%7Blog_b27%7D%7Blog_b16%7D%20%3D%20log_b27%20-%20log_b16%5C%5C%5C%5C%20%5Cfrac%7Blog_b27%7D%7Blog_b16%7D%20%3D%20log_b3%5E3%20-%20log_b2%5E4%5C%5C%5C%5CBy%5C%20rule%5C%202%5C%5C%5C%5C%20%5Cfrac%7Blog_b27%7D%7Blog_b16%7D%20%3D%203log_b3%20-%204log_b2%20%5C%5C%5C%5CFrom%5C%20%28i%29%5C%5C%5C%5C%5Cfrac%7Blog_b27%7D%7Blog_b16%7D%20%3D%20%203y%20-%204x)
Thus the given are evaluated in terms of x and y
Proportions are corresponding assuming they address a similar relationship. One method for checking whether two proportions are corresponding is to keep in touch with them as divisions and afterward decrease them. Assuming the decreased divisions are something similar, your proportions are relative.
Answer:
(2)
The increased amount is 9
Answer:
hello
Step-by-step explanation:
Your awnser should be c as i understood.
The answer is -8 the angles of the triangle are 180 so
180=42+x+76+x+78
180=2x+196
2x=-16
x=-8