Which equation is equivalent to $\log _3(x+5)=2$ ?
The given equation is \(\log_3(x+5) = 2\). We know that a logarithmic expression of the form \(\log_b(A) = C\) is equivalent to the exponential expression \(b^C = A\), provided that \(b > 0, b eq 1,\) and \(A > 0\). This definition allows us to transform logarithmic equations into algebraic equations that are often easier to solve.
In the equation \(\log_3(x+5) = 2\), we can identify the components based on the standard logarithmic form \(\log_b(A) = C\):
By substituting these values into the equivalent exponential form \(b^C = A\), we obtain the equation \(3^2 = x + 5\).
Now we compare our derived equation, \(3^2 = x + 5\), with the given options:
Our result perfectly matches the third option. Thus, the equation equivalent to \(\log_3(x+5) = 2\) is \(3^2 = x + 5\).
Logarithmic properties are fundamental in many real-world applications where quantities change exponentially, such as calculating the pH of a solution, measuring sound intensity in decibels, or analyzing the growth of populations or investments. Understanding how to convert logarithmic expressions to exponential ones is a crucial skill for solving these types of equations. For example, if you are determining the time \(t\) it takes for an investment to grow to a certain amount using the formula \(A = P(1+r)^t\), you would take the logarithm of both sides to isolate \(t\), then potentially convert back to exponential form for calculation.