Algebra - Solving Exponential Equations

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Section 6.3 : Solving Exponential Equations

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4. Solve the following equation.

\[{7^{4 - x}} = {7^{4x}}\]

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Start Solution

Recall the property that says if \({b^x} = {b^y}\) then \(x = y\). Since each exponential has the same base, 7 in this case, we can use this property to just set the exponents equal. Doing this gives,

\[4 - x = 4x\] Show Step 2

Now all we need to do is solve the equation from Step 1 and that is a simple linear equation. Here is the solution work.

\[\begin{align*}4 - x & = 4x\\ 4 & = 5x\hspace{0.25in} \to \hspace{0.25in}x = \frac{4}{5}\end{align*}\]

So, the solution to the equation is then : \(\require{bbox} \bbox[2pt,border:1px solid black]{{x = \frac{4}{5}}}\).