Question

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (x+2)=1$$

   Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state.
$$\log (x+2)=1$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 4, Problem 39

Instant Answer

Solved by Expert Kader Bayar

05/09/2020

Step 1

Since the base of the logarithm is not given, we assume it to be 10. So, the equation $\log (x+2)=1$ becomes $10^1 = x+2$. Show more…

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Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (x+2)=1$$

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Key Concepts

Extraneous Solutions

Extraneous solutions are results that emerge from the algebraic manipulation of equations but do not satisfy the original equation, often due to restrictions such as domain limitations. It is important to verify each solution by substituting it back into the original equation to ensure that it does not violate any conditions, such as having a non-positive argument in a logarithm.

Domain Restrictions of Logarithmic Functions

Any logarithmic expression log_b(x) is defined only if its argument x is positive. When solving logarithmic equations, it is essential to check that the solutions obtained also satisfy the condition that the original argument is greater than zero, as failing to do so can result in invalid, or extraneous, solutions.

Definition of Logarithms

This concept involves understanding that a logarithm is the inverse of exponentiation. In general, for positive numbers b (with b ? 1) and x, the equation log_b(x) = y is equivalent to b^y = x. This fundamental relationship allows one to convert logarithmic equations into exponential equations, which can be easier to solve.

Changing from Logarithmic to Exponential Form

When solving a logarithmic equation, converting it into exponential form is a crucial step. This process involves rewriting the logarithmic statement log_b(expression) = value as an exponential statement b^(value) = expression, thereby simplifying the process of isolating the variable.

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