Solve the systems in Exercises 79–80. log_y x=3, log_y (4x)=5

Verified step by step guidance

1. Start by understanding the given system of equations: ${\log }_{y}x=3$ and ${\log }_y(4x)=5$.
2. Rewrite the logarithmic equations in their exponential form: $x=y^3$ and $4x=y^5$.
3. Substitute $x=y^3$ into the second equation: $4(y^3)=y^5$.
4. Simplify the equation: $4y^3=y^5$.
5. Divide both sides by $y^3$ to solve for $y$: $4=y^2$.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve equations involving exponents. The expression log_y x = a means that y raised to the power of a equals x (y^a = x). Understanding how to manipulate and convert between logarithmic and exponential forms is essential for solving logarithmic equations.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can simplify calculations. It states that log_b a = log_k a / log_k b for any positive k. This is particularly useful when dealing with logarithms that are not easily computable in their original base, enabling the use of common or natural logarithms.

Properties of Logarithms

Properties of logarithms, such as the product, quotient, and power rules, are fundamental for simplifying logarithmic expressions. For example, log_y (ab) = log_y a + log_y b and log_y (a/b) = log_y a - log_y b. These properties help in breaking down complex logarithmic equations into simpler components, making it easier to solve for unknown variables.