Solve for $x$:
$81^{x+5}=3^{2 x}$
The problem asks us to solve the exponential equation \(81^{x+5} = 3^{2x}\) for the variable \(x\). To solve this, we need to express both sides of the equation using the same base. Since 81 is a power of 3, we can rewrite it as \(81 = 3^4\). By substituting this into the original equation, we can equate the exponents and solve the resulting linear equation.
We start with the given equation:
\[81^{x+5} = 3^{2x}\]
Since \(81 = 3^4\), we can substitute \(3^4\) for 81:
\[(3^4)^{x+5} = 3^{2x}\]
Using the power of a power rule \((a^m)^n = a^{mn}\), we simplify the left side:
\[3^{4(x+5)} = 3^{2x}\]
\[3^{4x + 20} = 3^{2x}\]
Since the bases on both sides are equal (base 3) and the function \(f(y) = 3^y\) is a one-to-one function, we can set the exponents equal to each other:
\[4x + 20 = 2x\]
To solve for \(x\), subtract \(2x\) from both sides:
\[2x + 20 = 0\]
Subtract 20 from both sides:
\[2x = -20\]
Divide by 2:
\[x = -10\]
Thus, the solution to the equation is \(x = -10\).
Exponential equations are used in various fields like finance and science. For example, if a population doubles every specific interval, the population \(P\) at time \(t\) can be modeled by an exponential equation like \(P = P_0 \times 2^{kt}\). Solving such equations helps in predicting future population sizes or calculating the time taken for an investment to grow to a certain amount when compound interest is involved. In our case, solving \(81^{x+5} = 3^{2x}\) illustrates the fundamental technique of using common bases to handle exponential growth or decay.