Question

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log x=0$$

   Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state.
$$\log x=0$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 4, Problem 35

Instant Answer

Solved by Expert Kader Bayar

05/09/2020

Step 1

The base of the logarithm is not written, but we know that it is a common logarithm, so its base is 10. Therefore, the equation $\log x=0$ can be rewritten as $10^0 = x$. Show more…

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Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log x=0$$

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Key Concepts

Logarithms

Logarithms are the inverse function of exponentiation. They answer the question of what exponent is needed for a given base to produce a certain number. This concept is foundational when solving equations where the variable is inside a logarithm, as it allows one to 'undo' the logarithmic operation by converting back to an exponential form.

Domain of Logarithmic Functions

A critical aspect of working with logarithms is ensuring that the argument of the logarithm is positive. This domain restriction (the input must be greater than zero) is essential to consider when solving logarithmic equations to avoid including extraneous solutions that do not satisfy the original conditions.

Exponential Conversion

Converting a logarithmic equation into its exponential form is an important method used to isolate the variable. This conversion leverages the definition of logarithms, where if log_b(a) = c, then a must equal b raised to the power of c. This process simplifies the equation, making it easier to solve for the variable.

Extraneous Solutions

Extraneous solutions are potential answers that arise during the process of solving an equation but do not satisfy the original equation, particularly its domain restrictions. It is essential to check each solution in the context of the original logarithmic equation to eliminate any that do not meet the necessary conditions, such as requiring the argument of the logarithm to be positive.

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