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Example 1: Solve the logarithmic equation.
ln(x) + 2 = - 3ln(x) + 10
Solution to example 1
- combine like terms
- divide both sides by 4.
- exponentiate both sides.
- inverse property of exponents and logs
- check:
Left Side of equation:
ln(e2) + 2
= 2 + 2
= 4
Right Side of equation:
- 3ln(e2) + 10
= -3(2) + 10
= 4
- conclusion: The solution to the above equation is
Example 2: Solve the exponential equation.
2ex + e-x = 3
Solution to example 2
- multiply all terms by ex
- use exponents properties to simplify.
note that.
- let u = ex and rewrite the equation in u
- rewrite the equation
- solve, for u, the above quadratic equation
- but u = ex
- solve, for x, the first of the above equations
- take logarithms of both sides
- which gives
- solve, for x, the second of the above equations
- take logarithms of both sides
- which gives
- check:
1st solution x = 0
Left Side of equation: 2e0+e0 = 2*1 + 1 =3
Right Side of Equation: 3
2nd solution x = -ln2
Left Side of equation: 2e-ln2 + e-(-ln2)
= 2/eln2 + eln2 =
= 2/2 + 2 = 3
Right Side of Equation: 3
conclusion: The solutions to the given equation are
x = 0 and x = -ln2.
Take a self test on exponential and logarithmic functions..
More topics to explore and Tests:
Experiment and Explore Mathematics: Tutorials and Problems
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