Question

MATH HELP!!! 100PTS!!!

Answer 1

Answer:

a)  u - v

b)  2v - 2

c)  3u + 3

d)  ¹/₂

Step-by-step explanation:

Given:

 $u=\log_{10}3$

 $v=\log_{10}5$

Part (a)

Rewrite 0.6 as a fraction:

$\implies \log_{10}(0.6)=\log_{10}\left(\dfrac{3}{5}\right)$

$\textsf{Apply the quotient log law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay:$

$\implies \log_{10}\left(\dfrac{3}{5}\right)=\log_{10}3-\log_{10}5$

Substitute the values of u and v:

$\implies \log_{10}3-\log_{10}5=u-v$

Part (b)

Rewrite 0.25 as 25/100:

$\implies \log_{10}(0.25)=\log_{10}\left(\dfrac{25}{100}\right)$

$\textsf{Apply the quotient log law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay$

$\implies \log_{10}\left(\dfrac{25}{100}\right)=\log_{10}(25)-\log_{10}(100)$

Rewrite 25 as 5² and 100 as 10²:

$\implies \log_{10}(25)-\log_{10}(100)=\log_{10}(5^2)-\log_{10}(10^2)$

$\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax$

$\implies \log_{10}(5^2)-\log_{10}(10^2)=2\log_{10}5-2\log_{10}10$

$\textsf{Apply the log law}: \quad \log_aa=1$

$\implies 2\log_{10}5-2\log_{10}10=2\log_{10}5-2(1)$

Substitute the value of v:

$\implies 2\log_{10}5-2(1)=2v-2$

Part (c)

Rewrite 27000 as 30³:

$\implies \log_{10}(27000)=\log_{10}(30^3)$

$\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax$

$\implies \log_{10}(30^3)=3\log_{10}(30)$

$\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay$

$\implies 3\log_{10}(30)=3\log_{10}(3)+3\log_{10}(10)$

$\textsf{Apply the log law}: \quad \log_aa=1$

$\implies 3\log_{10}(3)+3\log_{10}(10)=3\log_{10}(3)+3(1)$

Substitute the value of u:

$\implies 3\log_{10}(3)+3(1)=3u+3$

Part (d)

Rewrite √10 as $10^{\frac{1}{2}}$ :

$\implies \log_{10}(\sqrt{10})=\log_{10}(10^{\frac{1}{2}})$

$\textsf{Appy the Power log law}: \quad \log_ax^n=n\log_ax$

$\implies \log_{10}(10^{\frac{1}{2}})=\dfrac{1}{2}\log_{10}(10)$

$\textsf{Apply the log law}: \quad \log_aa=1$

$\implies \dfrac{1}{2}\log_{10}(10)=\dfrac{1}{2}(1)=\dfrac{1}{2}$