Problem 80 - Algebra and Trigonometry

Question

Solve the systems. $$ \left\\{\begin{array}{l} {\log x^{2}=y+3} \\ {\log x=y-1} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 10,000 and y = 5.

Step 1: Simplification of First Equation

Let's simplify our first equation. Remembering that \( \log b^{n} \) can be expressed as \( n \log b \), so, \( \log x^{2} = 2 \log x \). Hence, our first equation becomes \( 2 \log x = y + 3 \) or \( y = 2 \log x - 3 \).

Step 2: Substitution Into Second Equation

The strategy now would be to substitute y from our simplified first equation (Step 1) into our second equation. Doing so, we get \( \log x = [2 \log x - 3] - 1 \) or \( \log x = 2 \log x - 4 \). Now, isolate \( \log x \) to one side of the equation to get \( \log x = 4 \). Which implies that \( x = 10^{4} = 10,000 \), because we applied the property of logarithm that states \( \log_{a}{b} = c \) means \( a^{c} = b \).

Step 3: Solve for y

We can find y by substituting the value of x we found in Step 2 into either of original equations. Let's substitute x into the second equation for simplicity: \( \log x = y - 1 -> \log 10,000 = y - 1 -> 4 = y - 1 \). Therefore, \( y = 5 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are types of equations where the unknown variable appears as part of a logarithm. These equations can often be solved by using properties of logarithms to isolate the variable. In our exercise, we see the equations \( \log x^{2} = y + 3 \) and \( \log x = y - 1 \). To solve a logarithmic equation, you typically use properties such as the Power Rule, which states that \( \log_b{a^n} = n \log_b{a} \), to simplify the equation. This was applied in Step 1 to transform \( \log x^{2} \) into \( 2\log x \). It's crucial to remember that taking the log of both sides of an equation can help in situations where the variable is inside an exponent.

Another useful property is the Change of Base formula \( \log_b{a} = \frac{\log_c{a}}{\log_c{b}} \), where 'c' is any positive number (except for 1). Moreover, to solve for x, it's necessary to understand that \( a^{\log_a{b}} = b \) allows us to 'undo' a logarithm and solve for the variable in its exponent.

Systems of Equations

A system of equations is a set of equations with multiple variables. The goal is to find a solution that satisfies all equations simultaneously. The system in our exercise contains two logarithmic equations with two variables, x and y. There are various methods to solve such systems: Substitution, Elimination, and Graphical. We've used the Substitution method, efficiently leveraging the simplification from the properties of logarithms. After simplifying the first logarithmic equation, we substitute the expression for y from the first equation into the second equation. This process eventually leads to an equation with a single variable, allowing us to find a numerical solution for x, and subsequently for y. Solving systems of equations helps develop critical thinking, as it requires identifying the most efficient strategy to find the solution.

Properties of Logarithms

The properties of logarithms are essential tools in solving logarithmic equations and systems. They include the Power Rule, Product Rule \( \log_b{mn} = \log_b{m} + \log_b{n} \), and Quotient Rule \( \log_b{\frac{m}{n}} = \log_b{m} - \log_b{n} \). These properties can simplify complex expressions or isolate the variable in an equation. In the step-by-step solution, the Power Rule transforms \( \log x^2 \) into \( 2\log x \) by moving the exponent to the front of the logarithm.

Understanding these properties allows for the manipulation of the equations in ways that make it easier to solve for the unknown. For example, recognizing when to combine or break apart logs using these rules is fundamental in simplifying logarithmic expressions. It's also important to reverse these operations; for instance, converting a sum of logs into a single logarithm of a product, or vice versa, where applicable.

Exponential Functions

Exponential functions are closely related to logarithms. The function \( f(x) = a^x \) where 'a' is a positive real number, not equal to 1, is called an exponential function. In relation to logarithms, if \( b = a^c \) then \( \log_a{b} = c \). They describe growth or decay processes such as population growth, radioactive decay, and interest calculations.

In our exercise, once we have \( \log x = 4 \) from Step 2, we solve for x by raising 10 to the power of both sides, getting \( x = 10^4 = 10000 \). This is because 10 is the base of the common (base-10) logarithm when the base is not specified. Exponential functions and logarithms are inverses of each other, and this inverse relationship is fundamental to understanding how to solve logarithmic equations involving exponential expressions.