Solve the systems in Exercises 79–80. log x^2=y+3, log x=y−1

Verified step by step guidance

1. Start by setting the two given equations: $\log (x^2)=y+3$ and $\log (x)=y-1$.
2. Use the property of logarithms that states $\log (a^b)=b\cdot \log (a)$ to rewrite the first equation as $2\cdot \log (x)=y+3$.
3. Now you have two equations: $2\cdot \log (x)=y+3$ and $\log (x)=y-1$.
4. Substitute $y=\log (x)+1$ from the second equation into the first equation.
5. Solve the resulting equation for $x$ by isolating $\log (x)$ and then exponentiating to find $x$.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined as log_b(a) = c if and only if b^c = a. They are essential for solving equations involving logarithms, as they allow us to express relationships between variables in a multiplicative form. Understanding properties of logarithms, such as the product, quotient, and power rules, is crucial for manipulating and solving logarithmic equations.

Systems of Equations

A system of equations consists of two or more equations that share common variables. Solving such systems involves finding values for the variables that satisfy all equations simultaneously. Techniques for solving systems include substitution, elimination, and graphical methods. In this case, the system involves logarithmic equations, which may require specific strategies to isolate variables.

Exponential and Logarithmic Relationships

Exponential and logarithmic functions are closely related; specifically, if y = log_b(x), then x = b^y. This relationship is fundamental when solving logarithmic equations, as it allows us to convert between logarithmic and exponential forms. Recognizing how to manipulate these forms is key to solving the given system of equations effectively.