Solve each equation.
$7^{y-3}=50$

Verified step by step guidance

1. First, recognize that the equation is in the form of an exponential equation: \( 7^{y-3} = 50 \). Our goal is to solve for \( y \).
2. To solve for \( y \), take the natural logarithm (ln) of both sides of the equation to bring down the exponent: \( \ln(7^{y-3}) = \ln(50) \).
3. Apply the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \) to the left side: \( (y-3) \cdot \ln(7) = \ln(50) \).
4. Isolate \( y-3 \) by dividing both sides by \( \ln(7) \): \( y-3 = \frac{\ln(50)}{\ln(7)} \).
5. Finally, solve for \( y \) by adding 3 to both sides: \( y = \frac{\ln(50)}{\ln(7)} + 3 \).

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear in the exponent. To solve these equations, one often uses logarithms, which are the inverse operations of exponentiation. For example, in the equation 7^{y-3} = 50, we can take the logarithm of both sides to isolate the variable y.

Logarithms

Logarithms are mathematical functions that help to solve exponential equations by expressing the exponent as a product. The logarithm of a number is the exponent to which a base must be raised to produce that number. For instance, using the equation 7^{y-3} = 50, we can apply the logarithm base 7 to both sides to find y-3.

Properties of Logarithms

The properties of logarithms, such as the product, quotient, and power rules, are essential for simplifying logarithmic expressions. These properties allow us to manipulate logarithmic equations effectively. For example, the power rule states that log_b(a^c) = c * log_b(a), which can be used to simplify the equation after applying logarithms to both sides.