Algebra

1)    

Solve for x:
6 + (log(4 x - 3))/(log(10)) = 4

Subtract 6 from both sides:
(log(4 x - 3))/(log(10)) = -2

Multiply both sides by log(10):
log(4 x - 3) = -2 log(10)

-2 log(10) = log(1/10^2) = log(1/100):
log(4 x - 3) = log(1/100)

Cancel logarithms by taking exp of both sides:
4 x - 3 = 1/100

Add 3 to both sides:
4 x = 301/100

Divide both sides by 4:
Answer: |x = 301/400

1)$6 - \log \dfrac{1}{2} x = 3\\ \log \dfrac{1}{2} x = 6 - 3 = 3\\ \dfrac{1}{2}x=10^3\\ x = 2\times 10^3 = 2000$

2)$\log(4x-3)+6 = 4\\ \log(4x-3)=-2\\ 4x-3 = 10^{-2}=\dfrac{1}{100}\\ 4x = \dfrac{301}{100}\\ x=\dfrac{301}{400}$

3)$\dfrac{2}{3}\log(5x) = 2\\ \log(5x) = 3\\ 5x = 10^3 = 1000\\ x = 200$

4)$2\log(250x)-6 = 4\\ 2\log(250x) = 10\\ \log(250x)=5\\ 250x = 10^5 = 100000\\ x = 400$

5)$5-2\log x = \dfrac{1}{2}\\ 2\log x = 5-\dfrac{1}{2}=\dfrac{9}{2}\\ \log x = \dfrac{9}{4}\\ x = 10^{9/4}=\sqrt[4]{10^9}=100\cdot\sqrt[4]{10}$