# Math Help - Logarithms and surds

1. ## Logarithms and surds

Thanks for all your help. I appreciate it a lot.

Question 1
If $a^2 + b^2 = 7ab$ show that

$2 \log_{10} \frac{a+b}{3} = \log_{10} a + \log_{10} b$

Question 2
If $1 + \log_{a} (7x - 3a) = 2 \log_{a} x + \log_{a} 2$ find in terms of a the possible values of x

2. Originally Posted by ecogreen
Thanks for all your help. I appreciate it a lot.

Question 1
If $a^2 + b^2 = 7ab$ show that

$2 \log_{10} \frac{a+b}{3} = \log_{10} a + \log_{10} b$

Question 2
If $1 + \log_{a} (7x - 3a) = 2 \log_{a} x + \log_{a} 2$ find in terms of a the possible values of x
Q.1 $a^2 + b^2 = 7ab$

Add 2ab to both the sides. Then

$a^2 + b^2 + 2ab = 9ab$

$(a+b)^2 = 9ab$

$(\frac{a+b}{3})^2 = ab$

Now take log on both the side to the base 10.

Q.2

$1 + \log_{a} (7x - 3a) = 2 \log_{a} x + \log_{a} 2$

$\log_{a}(a) + \log_{a} (7x - 3a) = \log_{a}2x^2$

$\log_{a}a(7x - 3a) = \log_{a}2x^2$

$2x^2 - 7ax + 3a^2 = 0$

Now solve the quadratic to find x in terms of a.