Problem 39 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The solution to the log equation \(\log(x+2) = 1\) is \(x = 8\).

Step 1: Convert the logarithmic equation to an exponential equation

The logarithmic equation \(\log(x+2) = 1\) can be rewritten in exponential form as \(10^1 = x+2\). This makes use of the fact that logarithms with no base specified are assumed to be base 10.

Step 2: Solve the exponential equation for x

The equation \(10^1 = x+2\) simplifies to \(10 = x+2\). We can solve for x by subtracting 2 from both sides, yielding \(x = 8\).

Step 3: Check the solution

Now, we replace \(x\) with 8 in the original log equation to ensure that the solution is valid (not extraneous): \(\log(8+2) = \log10\) which simplifies to \(1 = 1\). This verifies that the solution is correct.

Key Concepts

Solving Equations

When faced with a logarithmic equation, the first step is to rewrite it in a form that is more straightforward to solve. Logarithmic equations often involve converting to a different mathematical form. The solution lies in transforming the logarithmic expression into an exponential equation. This allows us to eliminate the logarithm and work with a simple arithmetic equation instead.

The conversion uses the fundamental property of logarithms: if \( \log_b(a) = c \), then it can be expressed exponentially as \( b^c = a \). This is what we use when we have an unspecified logarithm base, which defaults to 10. Thus, solving the logarithmic equation \( \log(x+2) = 1 \) is an exercise in manipulating it into an exponential format, making it easier to solve for \( x \).

• First: Identify the logarithmic form.
• Second: Rewrite it in exponential form.
• Third: Solve the resulting equation.

The beauty of solving equations is that once you understand the conversion, the steps become much clearer and more systematic.

Exponential Form

Converting logarithmic expressions into exponential form is a key step in solving logarithmic equations. By applying the properties of logarithms, we replace the complex logarithmic format with a simpler arithmetic equation.

In the given exercise, the equation \( \log(x+2) = 1 \) implies a base 10 log because no base is specified. Hence, it can be rewritten exponentially as \( 10^1 = x+2 \). Here, we've transformed the problem from a logarithmic setting into an exponential one.

• The base is recognized, here as 10.
• The logarithm's result becomes the exponent.
• The number inside the logarithm is the result in the exponential form.

This exponential simplification makes the equation straightforward: simply solve \( 10 = x + 2 \). Subtracting 2 gives the clear solution \( x = 8 \). This transformation not only solves the problem but enhances our understanding of the strong link between logarithms and exponents.

Extraneous Solutions

Extraneous solutions may appear in the process of solving equations. They are solutions that emerge from the algebraic steps that do not actually satisfy the original equation. This concept is essential to validate the correctness of your answer.

For any equation, after performing operations and reaching a solution, it’s important to substitute back into the original equation. This ensures no extraneous results were introduced during transformations.

• Step 1: Solve the problem using proper methods, such as converting to exponential form.
• Step 2: Substitute the solution back into the initial equation.
• Step 3: Confirm if both sides of the equation hold true.

In this exercise, substituting \( x = 8 \) back into the original equation \( \log(x+2)=1 \) results in \( \log(10) = 1 \), confirming its correctness since \( 10^1 \) indeed equals 10. Thus, there are no extraneous solutions, making \( x = 8 \) the valid answer.