Problem 19 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=\log _{5} y $$

Step-by-Step Solution

Answer

The solution for \(y\) in terms of \(x\) is \(y = 5^x\).

Step 1: Identify the Equation Format

The equation given is in logarithmic form: \(x = \log_{5} y\). This identifies that you have a logarithm with base 5.

Step 2: Convert to Exponential Form

To solve for \(y\), convert the logarithmic equation to exponential form. From \(x = \log_{5} y\), we know that \(y = 5^x\). This conversion comes from the property that if \(a = \log_b c\), then \(c = b^a\).

Step 3: Verify the Expression for y

Double-check that the rewritten expression \(y = 5^x\) makes sense with the original problem statement. If \(x = \log_{5} y\), then it follows that \(y\) is the number such that base 5 raised to \(x\) equals \(y\). This confirms that \(y = 5^x\) is correct.

Key Concepts

Exponential Equations

Exponential equations are crucial in both mathematics and real-world applications. An exponential equation is one where a variable appears in the exponent. In the example given, we deal with converting a logarithmic equation to an exponential one to solve for a variable.
The equation \(x = \log_{5} y\) is initially in logarithmic form. The goal is to express \(y\) in terms of \(x\) using exponential forms. This is feasible because logarithms and exponents are inverse functions.
To transform a logarithmic equation \(x = \log_b y\) into an exponential equation, we rearrange it to \(y = b^x\). For the specific problem, we replaced the base \(b\) with 5, leading to \(y = 5^x\). This exponential equation expresses \(y\) as \(5\) raised to the power of \(x\). Understanding this transformation helps clarify how logs and exponents relate.

Logarithm Properties

Logarithm properties are essential tools that help in manipulating and solving equations involving logarithms. The primary property used in the solution is the inverse relationship between logarithms and exponents.
Let's delve into key properties:

• Inverse Property: If \(x = \log_b y\), then \(b^x = y\). This property demonstrates how a logarithm can be converted into an exponent, which is exactly what was performed in the exercise.
• Product Property: This states that \(\log_b (MN) = \log_b M + \log_b N\). Although not directly used here, it's a powerful tool in simplifying complex expressions.
• Quotient Property: This property expresses that \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). It’s handy for solving equations that involve division inside the log function.

Grasping these properties ensures you can fluidly navigate problems involving logarithms, transforming, and simplifying them as needed.

Base Conversion

Base conversion in logarithms allows altering the base of a logarithmic expression, making calculations more manageable or compatible with different systems.
In mathematics, it's common to encounter logs of bases different from what calculators or certain formulas accept. The change of base formula comes in handy:
For \(\log_b a\), convert it using:\[ \log_b a = \frac{\log_k a}{\log_k b} \]where \(k\) is the new base you want to use, often base 10 (\(\log\)) or base \(e\) (\(\ln\)).
Applying this, you could rewrite \(x = \log_5 y\) using base 10 logarithms for numerical calculations by:\[ x = \frac{\log_{10} y}{\log_{10} 5} \]Base conversion is potent for computational efficiency and standardizing complex equations across different bases.