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# 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