What are the steps in solving logarithmic equations and inequalities?

Solving Logarithmic Equations

1. Step 1: Use the rules of exponents to isolate a logarithmic expression (with the same base) on both sides of the equation.
2. Step 2: Set the arguments equal to each other.
3. Step 3: Solve the resulting equation.
4. Step 4: Check your answers.
5. Solve.

What is a logarithmic inequality?

Logarithmic inequalities are inequalities in which one (or both) sides involve a logarithm. Like exponential inequalities, they are useful in analyzing situations involving repeated multiplication, such as in the cases of interest and exponential decay.

What are the example of logarithmic inequalities?

The argument of a logarithm must be positive! Thus, it is also necessary to take into account any inequalities resulting from the arguments being positive; for example, an inequality involving the term log ⁡ 2 ( 2 x − 3 ) \log_2 (2x-3) log2​(2x−3) immediately requires x > 3 2 x>\frac{3}{2} x>23​.

Is an inequality involving logarithms?

How do you solve logarithmic equations with substitute solutions?

Remember to always substitute the possible solutions back to the original log equation. x = – 2 x = −2 if they will be valid solutions. Substitute back into the original logarithmic equation and verify if it yields a true statement. x = 5 x = 5 is definitely a solution. However, x =-2 x = −2 as part of our solution.

How to solve ln(x -3) x -2?

Solution: 1 Note the first term Ln ( x -3) is valid only when x >3; the term Ln ( x -2) is valid only when x >2; and the 2 Simplify the left side of the above equation: By the properties of logarithms, we know that 3 The equation can now be written 4 Let each side of the above equation be the exponent of the base e:

How do you work out logarithmic inequalities?

Introduction. The key to working with logarithmic inequalities is the following fact: If a > 1 and x > y, then logax > logay. Otherwise, if 0 < a < 1, then logax < logay. Of course, the base of a logarithm cannot be 1 or nonpositive. More importantly, the converse is true as well: If a > 1 and logax > logay, then x > y.

How do you solve a logarithmic equation with fractions?

Example 7: Solve the logarithmic equation. Collect all the logarithmic expressions on one side of the equation (keep it on the left) and move the constant to the right side. Use the Quotient Rule to express the difference of logs as fractions inside the parenthesis of the logarithm.