Question

Solve each system. Use any method you wish. $$ \left\\{\begin{array}{r} \log _{x} y=3 \\ \log _{x}(4 y)=5 \end{array}\right. $$

Short answer

The solution is \(x = 2\) and \(y = 8\).

Step-by-step solution

Express logarithmic equations as exponentials

Convert the logarithmic equations to their exponential form. From the first equation \(\text{log}_{x} y = 3\), the exponential form is \(y = x^3\). From the second equation \(\text{log}_{x}(4y) = 5\), the exponential form is \(4y = x^5\).

Substitute y from the first equation into the second equation

Substitute \(y = x^3\) into \(4y = x^5\) from the second equation. This results in \(4x^3 = x^5\).

Simplify the equation

Simplify \(4x^3 = x^5\) by dividing both sides by \(x^3\), provided \(x eq 0\). This gives \(4 = x^2\).

Solve for x

Solve the simplified equation \(4 = x^2\). Taking the square root of both sides, \(x = \pm2\). Since \(x\) must be a positive base of a logarithm, \(x = 2\).

Solve for y

Substitute \(x = 2\) back into the first exponential equation \(y = x^3\) to find \(y = 2^3 = 8\).

Key concepts

Logarithmic Equations

Logarithmic equations involve the logarithm of a variable set to a value. They are important in algebra and real-world applications.

A logarithm tells us how many times to multiply a base number to get another number. For example, in the equation \(\text{log}_b(a) = c\), it means that the base \(b\) raised to the power of \(c\) equals \(a\).

In this exercise, we start with the equations \( \text{log}_x y = 3 \) and \( \text{log}_x(4y) = 5 \). These equations mean we have an unknown base, \(x\), and we need to find its relationship with \(y\).

Understanding logarithmic equations is crucial for solving many mathematical problems. They can appear complex but become easier once we translate them into their exponential forms.

Exponential Form

To solve logarithmic equations, one useful method is converting them into their exponential form. This conversion makes the equations easier to handle with basic algebra.

The equation \( \text{log}_x y = 3 \) can be rewritten as \(y = x^3 \). Similarly, \( \text{log}_x(4y) = 5 \) converts to \(4y = x^5 \).

The conversion is done by remembering that \( \text{log}_x y = n \) implies \( x^n = y \). This shift can often make the solution more straightforward. Exponential forms help in equating the powers and simplifying the overall problem, making them a powerful tool in solving logarithmic equations.

Substitution Method

The substitution method is a powerful algebraic tool. This involves replacing one variable with another's known value.

After converting the logarithmic equations into \(y = x^3\) and \(4y = x^5\), we substitute \(y \) from the first into the second equation. Thus, we write \(4(x^3) = x^5\).

By doing this, we reduce the number of variables, making simpler equations to solve. This method is very effective when you have equations that can be easily manipulated to express one variable in terms of another.

Simplifying Equations

Simplifying equations is a critical topic in algebra. It involves reducing complex equations into simpler forms, making them easier to solve.

In the given problem, we start with \(4x^3 = x^5\). By dividing both sides by \(x^3\), we avoid dealing with higher powers of the variable. We get to \(4 = x^2\). Simplified equations are easier to solve and understand.

Always check for possible restrictions, such as \(x \eq 0\) here, to ensure valid operations. Finally, finding \(x = \pm2\) and selecting \(x = 2\) because logarithm bases must be positive, shows the importance of correct simplification in solving problems accurately.