Verified Answer :

  • Define the base, argument, and exponent of the given logarithm \(\log_4(8x) = 3\).
  • Convert the expression into the equivalent exponential form: \(4^3 = 8x\).
  • Simplify the equation to \(64 = 8x\) and divide by \(8\).
  • Determine the final value \(x = 8\).
  • Final answer: \(\boxed{8}\).

Explanation

  1. Analyzing the Logarithmic Equation

To solve the equation \(\log_4(8x) = 3\), we begin by identifying the components of the logarithmic expression. A logarithm of the form \(\log_b(A) = C\) states that the base \(b\) raised to the power of \(C\) equals the argument \(A\). In this specific case, the base \(b\) is \(4\), the argument \(A\) is \(8x\), and the exponent \(C\) is \(3\). We must also ensure that the argument \(8x\) is positive, which implies \(x > 0\).

  1. Rewriting and Evaluating the Equation

Using the definition of a logarithm, we can rewrite the equation in its exponential form:
\[4^3 = 8x\]
Calculating the value of \(4^3\), we get:
\[4 \times 4 \times 4 = 64\]
Substituting this value into the equation, we have:
\[64 = 8x\]

  1. Solving for x and Verification

To find the value of \(x\), we divide both sides of the equation by 8:
\[x = \frac{64}{8}\]
\[x = 8\]
Since \(x = 8\) is positive, it satisfies the condition for the argument of the logarithm to be valid. Therefore, the solution is \(x = 8\), which corresponds to option (b).

Examples

Logarithms are used in many real-world scenarios to handle quantities that span vast ranges, such as the pH level of a liquid, the intensity of earthquakes on the Richter scale, or the sound volume measured in decibels. For example, if a researcher knows the logarithmic relationship between the intensity of a sound and its decibel level, they can solve equations similar to this one to determine the actual intensity of the sound. This helps in understanding and managing environmental noise levels or calibrating acoustic equipment.

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