- Identify the domain: For \(\log_4(x+8)\) to be defined, we require \(x+8 > 0\), so \(x > -8\).
- Convert to exponential form: Use the property \(\log_b(A)=C \implies A=b^C\) to get \(x+8 = 4^2\).
- Simplify: Solve \(x+8 = 16\) to find \(x=8\).
- Verify: Confirm that \(8 > -8\), meaning the solution is valid.
- Final answer: \(\boxed{8}\)
Explanation
- Analyzing the Domain and Logarithmic Definition
To solve the equation \(\log_4(x+8) = 2\), we first identify the domain of the expression. The logarithmic function \(\log_b(A)\) is defined only when the argument \(A > 0\). In our case, the argument is \(x+8\), so we must have:
\[x + 8 > 0 \implies x > -8\]
This is the condition that any valid solution must satisfy.
- Solving the Equation in Exponential Form
The definition of a logarithm states that \(\log_b(A) = C\) is equivalent to the exponential form \(A = b^C\). Applying this definition to our equation \(\log_4(x+8) = 2\), we rewrite it as:
\[x + 8 = 4^2\]
Evaluating the power on the right side, we get:
\[x + 8 = 16\]
Subtracting 8 from both sides of the equation yields:
\[x = 16 - 8\]
\[x = 8\]
- Verifying the Solution
Finally, we must verify that our solution \(x = 8\) satisfies the domain constraint we identified in the first step. Since \(8 > -8\), the solution is valid and not extraneous. Thus, the final answer is \(x = 8\).
Examples
Logarithms are used in many real-world fields, such as measuring the intensity of earthquakes using the Richter scale, determining the pH level of chemical solutions, or calculating the time required for an investment to grow under compound interest. Solving equations like \(\log_4(x+8) = 2\) helps in understanding how input variables are related to these logarithmic scales, enabling precise predictions in science and finance.