Solve simple logarithmic equations

Solving simple logarithmic equations involves interpreting logarithms as exponents and using the inverse relationship between logarithmic and exponential forms to solve equations such as $\log_b(x)=c$ , $\log_b(f(x))=c$ , or equal logarithms with the same valid base. The reasoning includes enforcing positive arguments and rejecting solutions outside the domain; this scope excludes complex solutions, nonlinear logarithmic systems, and advanced manipulation of lengthy logarithmic expressions.

Detailed Explanation: Solve simple logarithmic equations

A logarithmic equation can often be solved by rewriting it in exponential form.

Remember:

\log_b(x)=c \quad \Longleftrightarrow \quad b^c=x

The base must satisfy $b>0$ and $b\ne 1$ , and the logarithm’s argument must be positive.

Example

Solve:

\log_3(2x-1)=2

Step 1: Rewrite in exponential form.

The base is $3$ , the exponent is $2$ , and the argument is $2x-1$ :

3^2=2x-1

Step 2: Simplify and solve for $x$ .

9=2x-1

Add $1$ to both sides:

10=2x

Divide by $2$ :

x=5

Step 3: Check the domain.

The argument of a logarithm must be positive:

2x-1>0

Substitute $x=5$ :

2(5)-1=9>0

So the solution is valid.

\boxed{x=5}

Learn by doing: Solve simple logarithmic equations

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Logarithms - Solve Log Equation (From Decimals)

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