# Solving logarithmic equations algebraically

1. $$\log_{10}(x+4) -\log_{10}x = \log_{10}(x+2)$$

$$10^{\log_{10}(x+4)} - 10^{\log_{10}x} = 10^{ \log_{10}(x+2)}$$

$$x+4 - x = x+2$$

$$x=2$$

2. $$\ln(x+1)^2 = 2$$

$$e^{\ln(x+1)^2 } = e^{2}$$

$$(x+1)^2 = e^2$$

$$x+1 = \pm e$$

$$x = -1\pm e$$

3. $$\ln x +\ln (x^2+1) = 8$$

$$x +(x^2+1) = e^8$$

can't factor this any further?

1. $$\log_{10}8x-\log_{10}(1+\sqrt{x}) =2$$

Putting everything to the 10th power gives

$$8x-(1+\sqrt x) = 100$$

and then solve from there?

5. $$\log_3 x +\log_3 (x^2 -8) = \log_3 8x$$

$$x + x^2 -8= 8x$$

$$x^2 -7x -8 = 0$$