Verified Answer :

  • Identify the common base of the exponential expressions as \(7\), where \(343 = 7^3\) and \(49 = 7^2\).
  • Rewrite the equation as \((7^3)^x = (7^2)^{4-x}\) and simplify the exponents using the power rule to obtain \(7^{3x} = 7^{8-2x}\).
  • Equate the exponents to form the linear equation \(3x = 8-2x\).
  • Solve the linear equation to find \(x = \frac{8}{5}\), which corresponds to option d.
  • The final answer is \(\boxed{\frac{8}{5}}\).

Explanation

  1. Analyzing the Problem and Simplifying the Equation

The goal is to solve for \(x\) in the exponential equation \(343^x = 49^{4-x}\). Both sides of the equation involve powers of base 7. We recall that \(343 = 7^3\) and \(49 = 7^2\). By expressing both sides with the same base, we can simplify the equation into a linear one by equating the exponents.

  1. Applying Exponent Laws and Equating Exponents

Substitute \(343 = 7^3\) and \(49 = 7^2\) into the original equation:
\[(7^3)^x = (7^2)^{4-x}\]
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we simplify both sides:
\[7^{3x} = 7^{2(4-x)}\]
Since the bases are equal, we equate the exponents:
\[3x = 2(4-x)\]

  1. Solving for x

Now, solve the linear equation for \(x\):
\[3x = 8 - 2x\]
Add \(2x\) to both sides of the equation:
\[3x + 2x = 8\]
\[5x = 8\]
Divide by 5:
\[x = \frac{8}{5}\]
Comparing this result to the given options, we find that it matches option d.

Examples

Exponential equations like this are common in finance and science. For instance, if you are modeling the growth of an investment or the decay of a substance, you often equate exponential expressions to find the time \(x\) at which two different quantities become equal. This specific problem demonstrates the principle of using common bases to solve for an unknown variable in the exponent.

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