Problem 13 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=8^{y} $$

Step-by-Step Solution

Answer

\( y = \frac{\ln(x)}{\ln(8)} \)

Step 1: Understand the Equation

The given equation is \( x = 8^y \). We need to express \( y \) in terms of \( x \), meaning \( y \) must be on one side of the equation and \( x \) on the other side.

Step 2: Apply the Logarithmic Function

To solve for \( y \), apply the natural logarithm (\( \ln \)) to both sides of the equation. This is because logarithms are the inverse operations of exponentiation: \( \ln(x) = \ln(8^y) \).

Step 3: Use Logarithmic Properties

Utilize the property of logarithms that allows exponents to be brought down as coefficients: \( \ln(8^y) = y \ln(8) \). Thus, the equation becomes \( \ln(x) = y \ln(8) \).

Step 4: Solve for y

To solve for \( y \), divide both sides by \( \ln(8) \): \[ y = \frac{\ln(x)}{\ln(8)} \]. Now \( y \) is expressed in terms of \( x \).

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential tool for solving exponential equations. When you have an equation where the variable is an exponent, logarithms can help simplify and solve it. The core idea is that a logarithm finds the power to which a base number must be raised to achieve a given value. For example, the logarithm base 8 of a number is the power to which 8 must be raised to get that number.

In mathematical notation, the logarithm of a number can be represented in different bases, but the most common are base 10, written as \( \log \), and the natural base (approximately 2.718), written as \( \ln \).

For our exercise, we used the natural logarithm, \( \ln \), to manage the exponential equation. By applying the logarithmic function to both sides of the exponential equation \( x = 8^y \), we start converting the equation into a form that's easier to manipulate and solve.

Inverse Operations

Inverse operations are operations that "undo" each other. When you deal with exponential equations, logarithms serve as the inverse operation to exponentiation.

This relationship is a fundamental aspect to grasp because it allows us to reverse the process of raising a number to a power. In the context of solving the equation \( x = 8^y \), exponentiation is the initial operation applied to \( y \). Therefore, taking the logarithm of both sides effectively "undoes" this exponentiation, bringing us closer to isolating \( y \).

Using inverse operations simplifies problems that can otherwise seem complex. You'll often find that understanding and mastering this concept will make logarithms and exponential functions a lot more manageable.

Logarithm Properties

Logarithm properties are rules and identities that simplify computations involving logarithms. One key property is that the logarithm of a power allows you to bring the exponent down as a coefficient. This property is crucial when solving exponential equations.

For our equation, \( x = 8^y \), we applied the property that \( \ln(8^y) = y \ln(8) \). Bringing \( y \) down as a coefficient transforms the equation into something much more manageable. It moves from an exponential context, where \( y \) is trapped in the exponent, to a linear one, allowing for direct algebraic manipulation.

Another important property is the change of base formula, which can help when dealing with different logarithmic bases. Understanding these properties not only aids in solving problems but also enriches your overall mathematical toolkit.