# Logarithms Equation Example

- Question:
- Given: lg(x + 3)
^{3} - lg^{2}(x + 3) - 2 = 0, What is the value of x?

- Solution:
- lg(x + 3)
^{3} - lg^{2}(x + 3) - 2 = 0
- lg
^{2}(x + 3) - lg(x + 3)^{3} + 2 = 0
- lg
^{2}(x + 3) - 3 lg(x + 3) + 2 = 0
- let t = lg(x + 3), then lg
^{2}(x + 3) - 3 lg(x + 3) + 2 = 0 change to:
- t
^{2} - 3t + 2 = 0
- (t - 2) (t - 1) = 0
- t
_{1} = 2 and t_{2} = 1
- substitute the value of t to the equation t = lg(x + 3)
- equation1:
- 2 = lg(x + 3)
- 10
^{2} = x + 3
- x = 10
^{2} - 3
- x = 97
- equation2:
- 1 = lg(x + 3)
- 10
^{1} = x + 3
- x = 10 - 3
- x = 7
- The solution is: x
_{1} = 7 and x_{2} = 97

- Formula used in this example:
- log
_{a}b^{x} = x log_{a}b
- If a
^{x} = b (a > 0 and a is not equal to 1), then x = log_{a}b
- when a = 10, log
_{10}b = lg b