Question

Solve each logarithmic equation. $$\log _{x} 4=2$$

   Solve each logarithmic equation.
$$\log _{x} 4=2$$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 2nd Edition

Chapter 13, Problem 40

Instant Answer

Solved by Expert Liuxi Sun

06/26/2020

Step 1

We can convert this logarithmic equation into an exponential equation. The base is $x$, the exponent is $2$ and the result is $4$. So, we can write this as $x^2 = 4$. Show more…

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Solve each logarithmic equation. $$\log _{x} 4=2$$

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Key Concepts

Definition of Logarithms

The definition of a logarithm states that log base b of a equals c if and only if b raised to the power c is equal to a. This principle is fundamental in understanding how logarithmic equations are structured and forms the basis for converting between logarithmic and exponential forms.

Conversion to Exponential Form

Solving logarithmic equations often involves rewriting the logarithmic expression as an equivalent exponential equation. Using the definition, an equation of the form log_b(a) = c can be rewritten as b^c = a, which simplifies the process of isolating and solving for the variable in the equation.

Domain Restrictions

Logarithms come with specific domain restrictions: the base of a logarithm must be positive and not equal to 1, and the argument of the logarithm must be positive. These restrictions are crucial because they ensure the logarithm is well-defined and the solutions obtained are valid within the context of the equation.

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