Skill: Solve simple logarithmic equations

Explanation and Free Practice Resources

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.

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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}

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Logarithms - Solve Log Equation (From Decimals)

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