# Logarithms and surds

#### ecogreen

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

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

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

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

#### sa-ri-ga-ma

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

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

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

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

Add 2ab to both the sides. Then

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

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

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

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

Q.2

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

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

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

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

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

• ecogreen