Verified Answer :

  • Rewrite the logarithmic equation in its equivalent exponential form: \(4^2 = 7t + 2\).
  • Simplify and solve the resulting linear equation for \(t\): \(16 = 7t + 2 \implies 14 = 7t \implies t = 2\).
  • Confirm the validity of the solution by checking the domain of the logarithm: \(7(2) + 2 = 16 > 0\).
  • The final answer is \(\boxed{2}\).

Explanation

  1. Using the Definition of a Logarithm

To solve the equation \(\log_{4}(7t + 2) = 2\), we start by using the definition of a logarithm. A logarithmic equation of the form \(\log_b(A) = C\) is equivalent to the exponential form \(b^C = A\). Here, the base \(b\) is 4, the argument \(A\) is \(7t + 2\), and the result \(C\) is 2. Therefore, we can rewrite the equation as \(4^2 = 7t + 2\).

  1. Solving the Resulting Linear Equation

Next, we evaluate the exponent: \(4^2 = 16\). Substituting this back into our equation, we get:
\[16 = 7t + 2\]
To isolate the term with \(t\), we subtract 2 from both sides of the equation:
\[16 - 2 = 7t\]
\[14 = 7t\]
Now, divide both sides by 7 to solve for \(t\):
\[t = \frac{14}{7}\]
\[t = 2\]

  1. Verifying the Solution

Finally, we must verify that our solution is valid within the domain of the logarithm. A logarithmic expression \(\log_b(x)\) is only defined for \(x > 0\). In our case, the argument is \(7t + 2\). Substituting \(t = 2\), we have \(7(2) + 2 = 14 + 2 = 16\). Since \(16 > 0\), the solution \(t = 2\) is valid. Thus, the final value for \(t\) is 2.

Examples

Logarithms are used in many scientific fields to handle wide ranges of values. For instance, the Richter scale for measuring earthquake magnitude is logarithmic; an increase of 1 unit on the scale represents a tenfold increase in the measured amplitude of seismic waves. Similarly, in chemistry, the pH of a solution, which measures acidity, is calculated as the negative logarithm of the hydrogen ion concentration. Solving logarithmic equations like the one in this problem helps professionals determine unknown concentrations or magnitudes when given a measured output, essentially reversing the logarithmic scale to find the original input.

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