If logb2=x and logb3=y, evaluate the following in terms of x and y:
1 answer:
![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
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