Problem 11 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=6^{y} $$

Step-by-Step Solution

Answer

\(y = \frac{\log(x)}{\log(6)}\)

Step 1: Introduction to the Equation Type

The equation given is an exponential equation where the variable \(y\) is in the exponent: \(x = 6^y\). Our goal is to express \(y\) in terms of \(x\).

Step 2: Applying Logarithms

To solve for \(y\), we need to eliminate the exponent by applying logarithms. Use logarithm properties to both sides of the equation: \(\log(x) = \log(6^y)\).

Step 3: Using Logarithm Power Rule

According to the power rule of logarithms, \(\log(a^b) = b \cdot \log(a)\). Apply this to the equation: \(\log(x) = y \cdot \log(6)\).

Step 4: Isolating y

To solve for \(y\), divide both sides by \(\log(6)\): \(y = \frac{\log(x)}{\log(6)}\).

Step 5: Final Answer

The equation is solved for \(y\) in terms of \(x\): \(y = \frac{\log(x)}{\log(6)}\).

Key Concepts

Logarithms

Logarithms are a concept that reverses the operation of exponentiation. If you have an equation where a number is raised to a power, like in an exponential equation, logarithms help us find the unknown exponent. Imagine a simple equation like \(x = b^y\), where \(b\) is the base, and \(y\) is the exponent – a logarithm can help isolate \(y\). To rewrite this equation in terms of logarithms, you would write \(y = \log_b(x)\), where \(\log_b\) denotes the logarithm base \(b\). This is a powerful tool to convert an exponential equation into a linear form, making the task of isolating a variable much simpler.

• Logarithms answer the question: "To what power must the base be raised, to get the given number?"
• They are written as \(\log(x)\), where \(x\) is the number you're evaluating.
• Common bases for logarithms are 10 (common logarithm) and \(e\) (natural logarithm).

Understanding logarithms begins with recognizing their relationship with exponents. In solving equations, they are invaluable for simplifying expressions where the unknown variable is an exponent.

Logarithm Power Rule

The logarithm power rule is an essential property used when dealing with logarithmic expressions involving exponents. The rule states that \(\log(a^b) = b \cdot \log(a)\). This property is instrumental in solving equations where variables are in the exponent, like in our problem \(x = 6^y\). By applying the logarithm power rule, you can simplify and manage complex exponential equations.

Consider the equation \(\log(6^y)\). Using the power rule, this simplifies to \(y \cdot \log(6)\). This crucial step assists in the isolation of variables, allowing us to break down expressions that initially appear intimidating.

• It converts the exponential form into a multiplication form, easy to manipulate.
• Helps in directly extracting exponents from within logarithmic expressions.
• Simplifies equations and makes algebraic manipulation possible.

The logarithm power rule is not just useful; it's necessary for transitioning between exponential and linear forms. This conversion is what enables us to solve for variables effectively.

Solving for a Variable

When solving for a variable in an equation, our goal is to isolate the variable on one side, usually to express it in terms of other known quantities. In our previous example \(x = 6^y\), \(y\) is the variable we want to solve for. We approach this systematically using various mathematical tools, such as logarithms, to transition from a complex form into a simpler one.

In the process of solving, the following steps can be employed:

• Convert Exponentials to Logarithms: By applying logarithms to both sides, the complex equation takes on a linear form.
• Apply Properties: Use properties like the logarithm power rule to further simplify.
• Isolate the Variable: Use algebraic operations to solve for the variable, typically involving simplifying fractions or multiplying/dividing.

In our equation, applying logarithms helps transform \(6^y\) into \(y \cdot \log(6)\). From there, isolating \(y\) involves dividing both sides by \(\log(6)\), leaving \(y = \frac{\log(x)}{\log(6)}\). This step-by-step breakdown demystifies the isolation of variables and showcases the power of logarithms in simplifying exponential equations.