Solve Exponential and Logarithmic Equations  Tutorial
How to solve exponential and logarithmic equations? A tutorial with examples and detailed solutions. Logarithmic equations are equations with logarithmic terms and exponential equations are equations with exponential terms.
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(e^{2}) + 2
= 2 + 2
= 4
Right Side of equation:
 3ln(e^{2}) + 10
= 3(2) + 10
= 4

conclusion: The solution to the above equation is
Example 2: Solve the exponential equation.
2e^{x} + e^{x} = 3
Solution to example 2

multiply all terms by e^{x}
(2e^{x} + e^{x})e^{x} = 3e^{x}

use exponents properties to simplify.

note that.

let u = e^{x} and rewrite the equation in u

rewrite the equation

solve, for u, the above quadratic equation

but u = e^{x}

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: 2e^{0}+e^{0} = 2*1 + 1 =3
Right Side of Equation: 3
2nd solution x = ln2
Left Side of equation: 2e^{ln2} + e^{(ln2)}
= 2/e^{ln2} + e^{ln2} =
= 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..
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