Verified Answer :

  • Determine the domain for the equation: \(\log_4(x+3) + \log_4 x = 1\) requires \(x > 0\).
  • Simplify the equation using logarithmic properties to \(\log_4(x^2 + 3x) = 1\).
  • Solve the resulting quadratic equation \(x^2 + 3x - 4 = 0\), yielding potential solutions \(x = -4\) and \(x = 1\).
  • Validate the solutions against the domain constraint \(x > 0\), confirming that only \(x = 1\) is a valid solution.
  • Final Answer: \(\boxed{x=1}\)

Explanation

  1. Identifying the Domain

The equation is \(\log_4(x+3) + \log_4 x = 1\). Before solving, we must identify the valid domain of \(x\). The argument of a logarithm must be strictly positive. Thus, we have two conditions:

  1. \(x+3 > 0 \implies x > -3\)
  2. \(x > 0 \implies x > 0\)

The intersection of these two conditions gives us the valid domain: \(x > 0\). Any potential solutions outside this range must be rejected.

  1. Combining Logarithms

We use the logarithmic identity \(\log_b(M) + \log_b(N) = \log_b(M \cdot N)\) to combine the terms on the left side:
\[\log_4((x+3) \cdot x) = 1\]
\[\log_4(x^2 + 3x) = 1\]

  1. Converting to an Exponential Equation

Using the definition of a logarithm, where \(\log_b(A) = C\) is equivalent to \(A = b^C\), we rewrite the equation in exponential form:
\[x^2 + 3x = 4^1\]
\[x^2 + 3x = 4\]
Subtracting 4 from both sides results in the quadratic equation:
\[x^2 + 3x - 4 = 0\]

  1. Solving the Quadratic Equation

To solve the quadratic equation \(x^2 + 3x - 4 = 0\), we look for two numbers that multiply to \(-4\) and add to \(3\). These numbers are \(4\) and \(-1\). Thus, we can factor the equation as:
\[(x+4)(x-1) = 0\]
This gives us two potential solutions: \(x = -4\) and \(x = 1\).

  1. Verifying Solutions and Selecting the Correct Answer

Finally, we check our potential solutions against the domain constraint established in step 1 (\(x > 0\)):

  • For \(x = -4\): \(-4 gtr 0\), so this solution is invalid.
  • For \(x = 1\): \(1 > 0\), so this solution is valid.

Therefore, the only valid solution is \(x = 1\). Comparing this to the given options, the correct answer is \(x=1\).

Examples

Logarithms are used to model phenomena that span multiple orders of magnitude. For instance, in finance, solving logarithmic equations helps determine the time needed for an investment to grow to a certain amount, given a compounding interest rate. If an investment's value follows a logarithmic growth curve, identifying the exact time \(x\) when it hits a target value involves solving equations similar to the one we just handled, ensuring the result is meaningful within the growth timeframe.

Related Questions