Solve logarithmic equations

Logarithmic equations are interpreted through the inverse relationship between logarithmic and exponential forms, with bases greater than 0 and not equal to 1, and logarithm arguments restricted to positive values. Solutions may involve isolating a logarithm, applying product, quotient, and power properties, or substituting to solve equations such as quadratic expressions in a logarithm; domain restrictions and verification are essential because algebraic transformations can produce extraneous solutions.

Detailed Explanation: Solve logarithmic equations

To solve a logarithmic equation:

State the domain: Every logarithm’s argument must be positive.
Use logarithm properties to combine or simplify logarithms.
Rewrite in exponential form or solve the resulting algebraic equation.
Check all answers in the original equation, because algebra can produce values outside the logarithm’s domain.

Example

Solve:

\log_2(x)+\log_2(x-2)=3

Step 1: Find the domain.

Both logarithm arguments must be positive:

x>0 \qquad \text{and} \qquad x-2>0

Together, these give:

x>2

Step 2: Combine the logarithms.

Using the product property,

\log_b M+\log_b N=\log_b(MN)

we get:

\log_2\bigl(x(x-2)\bigr)=3

Step 3: Rewrite in exponential form.

Since $\log_b A=c$ means $A=b^c$ ,

x(x-2)=2^3

So:

x^2-2x=8

x^2-2x-8=0

Step 4: Solve the quadratic.

Factor:

(x-4)(x+2)=0

Therefore,

x=4 \qquad \text{or} \qquad x=-2

Step 5: Check the domain.

The domain requires $x>2$ . Thus, $x=-2$ is not allowed.

Check $x=4$ in the original equation:

\log_2(4)+\log_2(2)=2+1=3

Therefore, the solution is:

\boxed{x=4}

Learn by doing: Solve logarithmic equations

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Algebra with Logarithms - Binomial over Binomial and Constant

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