They are 80km apart from each other
Answer:
∠3 = 81
Step-by-step explanation:
Interior angles in a triangle add up to equal 180°
Hence, 58° + 41° + ∠3 = 180
58 + 41 = 99
99 + ∠3 = 180
* subtract 99 from each side *
99 - 99 cancels out
180 - 99 = 81
we're left with ∠3 = 81°
Answer:
51 is the answer to the question
Answer:
a) u - v
b) 2v - 2
c) 3u + 3
d) ¹/₂
Step-by-step explanation:
Given:
![u=\log_{10}3](https://tex.z-dn.net/?f=u%3D%5Clog_%7B10%7D3)
![v=\log_{10}5](https://tex.z-dn.net/?f=v%3D%5Clog_%7B10%7D5)
<u>Part (a)</u>
Rewrite 0.6 as a fraction:
![\implies \log_{10}(0.6)=\log_{10}\left(\dfrac{3}{5}\right)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%280.6%29%3D%5Clog_%7B10%7D%5Cleft%28%5Cdfrac%7B3%7D%7B5%7D%5Cright%29)
![\textsf{Apply the quotient log law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay:](https://tex.z-dn.net/?f=%5Ctextsf%7BApply%20the%20quotient%20log%20law%7D%3A%20%5Cquad%20%5Clog_a%5Cfrac%7Bx%7D%7By%7D%3D%5Clog_ax%20-%20%5Clog_ay%3A)
![\implies \log_{10}\left(\dfrac{3}{5}\right)=\log_{10}3-\log_{10}5](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%5Cleft%28%5Cdfrac%7B3%7D%7B5%7D%5Cright%29%3D%5Clog_%7B10%7D3-%5Clog_%7B10%7D5)
Substitute the values of u and v:
![\implies \log_{10}3-\log_{10}5=u-v](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D3-%5Clog_%7B10%7D5%3Du-v)
<u>Part (b)</u>
Rewrite 0.25 as 25/100:
![\implies \log_{10}(0.25)=\log_{10}\left(\dfrac{25}{100}\right)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%280.25%29%3D%5Clog_%7B10%7D%5Cleft%28%5Cdfrac%7B25%7D%7B100%7D%5Cright%29)
![\textsf{Apply the quotient log law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay](https://tex.z-dn.net/?f=%5Ctextsf%7BApply%20the%20quotient%20log%20law%7D%3A%20%5Cquad%20%5Clog_a%5Cfrac%7Bx%7D%7By%7D%3D%5Clog_ax%20-%20%5Clog_ay)
![\implies \log_{10}\left(\dfrac{25}{100}\right)=\log_{10}(25)-\log_{10}(100)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%5Cleft%28%5Cdfrac%7B25%7D%7B100%7D%5Cright%29%3D%5Clog_%7B10%7D%2825%29-%5Clog_%7B10%7D%28100%29)
Rewrite 25 as 5² and 100 as 10²:
![\implies \log_{10}(25)-\log_{10}(100)=\log_{10}(5^2)-\log_{10}(10^2)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%2825%29-%5Clog_%7B10%7D%28100%29%3D%5Clog_%7B10%7D%285%5E2%29-%5Clog_%7B10%7D%2810%5E2%29)
![\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax](https://tex.z-dn.net/?f=%5Ctextsf%7BAppy%20the%20Power%20log%20law%7D%3A%20%5Cquad%20%5Clog_ax%5En%3Dn%5Clog_ax)
![\implies \log_{10}(5^2)-\log_{10}(10^2)=2\log_{10}5-2\log_{10}10](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%285%5E2%29-%5Clog_%7B10%7D%2810%5E2%29%3D2%5Clog_%7B10%7D5-2%5Clog_%7B10%7D10)
![\textsf{Apply the log law}: \quad \log_aa=1](https://tex.z-dn.net/?f=%5Ctextsf%7BApply%20the%20log%20law%7D%3A%20%5Cquad%20%5Clog_aa%3D1)
![\implies 2\log_{10}5-2\log_{10}10=2\log_{10}5-2(1)](https://tex.z-dn.net/?f=%5Cimplies%202%5Clog_%7B10%7D5-2%5Clog_%7B10%7D10%3D2%5Clog_%7B10%7D5-2%281%29)
Substitute the value of v:
![\implies 2\log_{10}5-2(1)=2v-2](https://tex.z-dn.net/?f=%5Cimplies%202%5Clog_%7B10%7D5-2%281%29%3D2v-2)
<u>Part (c)</u>
Rewrite 27000 as 30³:
![\implies \log_{10}(27000)=\log_{10}(30^3)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%2827000%29%3D%5Clog_%7B10%7D%2830%5E3%29)
![\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax](https://tex.z-dn.net/?f=%5Ctextsf%7BAppy%20the%20Power%20log%20law%7D%3A%20%5Cquad%20%5Clog_ax%5En%3Dn%5Clog_ax)
![\implies \log_{10}(30^3)=3\log_{10}(30)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%2830%5E3%29%3D3%5Clog_%7B10%7D%2830%29)
![\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay](https://tex.z-dn.net/?f=%5Ctextsf%7BApply%20the%20log%20product%20law%7D%3A%20%5Cquad%20%5Clog_axy%3D%5Clog_ax%20%2B%20%5Clog_ay)
![\implies 3\log_{10}(30)=3\log_{10}(3)+3\log_{10}(10)](https://tex.z-dn.net/?f=%5Cimplies%203%5Clog_%7B10%7D%2830%29%3D3%5Clog_%7B10%7D%283%29%2B3%5Clog_%7B10%7D%2810%29)
![\textsf{Apply the log law}: \quad \log_aa=1](https://tex.z-dn.net/?f=%5Ctextsf%7BApply%20the%20log%20law%7D%3A%20%5Cquad%20%5Clog_aa%3D1)
![\implies 3\log_{10}(3)+3\log_{10}(10)=3\log_{10}(3)+3(1)](https://tex.z-dn.net/?f=%5Cimplies%203%5Clog_%7B10%7D%283%29%2B3%5Clog_%7B10%7D%2810%29%3D3%5Clog_%7B10%7D%283%29%2B3%281%29)
Substitute the value of u:
![\implies 3\log_{10}(3)+3(1)=3u+3](https://tex.z-dn.net/?f=%5Cimplies%203%5Clog_%7B10%7D%283%29%2B3%281%29%3D3u%2B3)
<u>Part (d)</u>
Rewrite √10 as
:
![\implies \log_{10}(\sqrt{10})=\log_{10}(10^{\frac{1}{2}})](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%28%5Csqrt%7B10%7D%29%3D%5Clog_%7B10%7D%2810%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%29)
![\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax](https://tex.z-dn.net/?f=%5Ctextsf%7BAppy%20the%20Power%20log%20law%7D%3A%20%5Cquad%20%5Clog_ax%5En%3Dn%5Clog_ax)
![\implies \log_{10}(10^{\frac{1}{2}})=\dfrac{1}{2}\log_{10}(10)](https://tex.z-dn.net/?f=%5Cimplies%20%5Clog_%7B10%7D%2810%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%29%3D%5Cdfrac%7B1%7D%7B2%7D%5Clog_%7B10%7D%2810%29)
![\textsf{Apply the log law}: \quad \log_aa=1](https://tex.z-dn.net/?f=%5Ctextsf%7BApply%20the%20log%20law%7D%3A%20%5Cquad%20%5Clog_aa%3D1)
![\implies \dfrac{1}{2}\log_{10}(10)=\dfrac{1}{2}(1)=\dfrac{1}{2}](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7B1%7D%7B2%7D%5Clog_%7B10%7D%2810%29%3D%5Cdfrac%7B1%7D%7B2%7D%281%29%3D%5Cdfrac%7B1%7D%7B2%7D)
X - y = 4
-y = 4 + x
y = -4 -x
x + y = 4
So, it's B