Problem 9 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{e} \frac{1}{e^{2}}=-2 $$

Step-by-Step Solution

Answer

The exponential equation is \( e^{-2} = \frac{1}{e^{2}} \).

Step 1: Identify the Base of the Logarithm

The logarithm given is \( \log_{e} \), which means the base is \( e \). This is the natural logarithm, commonly denoted as \( \ln \).

Step 2: Understand the Logarithmic Equation

The equation given is \( \log _{e} \frac{1}{e^{2}}=-2 \). This says: what power must \( e \) be raised to, in order to get \( \frac{1}{e^{2}} \)? The answer is \(-2\).

Step 3: Convert the Logarithmic Equation to an Exponential Equation

To convert from a logarithmic form to exponential form, use the formula: if \( \log_{b}(x) = y \), then \( b^y = x \). Applying this, we have \( e^{-2} = \frac{1}{e^{2}} \).

Step 4: Validate the Exponential Form

Checking our work: raising \( e \) to the power of \(-2\) gives \( \frac{1}{e^{2}} \), which matches what we started with. Therefore, \( e^{-2} = \frac{1}{e^{2}} \) is the correct exponential form.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical sentences that involve logarithms. Here, the idea is to solve for a variable or validate the given equation by converting it back to its exponential form. The logarithm, denoted in various bases such as base 10 (common log) or base \(e\) (natural log), tells us the power to which the base must be raised to obtain a certain number. For example, in a logarithmic equation like \(\log_{b}(x) = y\), if the base \(b\) raised to the power \(y\) results in \(x\), then the equation holds true. Logarithmic equations are extensively used to simplify calculations involving exponential growth or decay, making them crucial in fields like science and engineering.

Natural Logarithms

Natural logarithms are a special type of logarithms where the base is the mathematical constant \(e\), approximately equal to 2.71828. They are written as \(\ln(x)\) instead of \(\log_{e}(x)\). Natural logarithms are particularly useful in calculus and complex exponential function analysis, where the derivative of \(\ln(x)\) is simply \(1/x\). This property makes them integral to solving equations involving growth processes such as population dynamics and radioactive decay.

• Natural logarithms simplify many mathematical models.
• The base \(e\) is unique because of its continuous growth rate.
• It simplifies integration and differentiation in calculus.

Converting Forms

Converting between forms, specifically switching from a logarithmic equation to an exponential form, is a fundamental skill. It involves using the relationship \(\log_{b}(x) = y\) to reformulate as \(b^{y} = x\). For instance, if we have \(\log_{e} \frac{1}{e^{2}} = -2\), this converts to the exponential form \(e^{-2} = \frac{1}{e^{2}}\). This transformation is essential since exponential equations are often easier to solve or interpret, especially when checking the validity of solutions.

• Identify the base and the result from the logarithm.
• Apply the conversion formula precisely.
• Verify by reconnecting exponential results with the original logarithmic idea.

Exponential Form

Exponential form of equations presents them as powers of a common base. When a logarithmic equation is given, converting it to an exponential form can offer clarity. For instance, the problem \(\log_{e}(x) = y\) becomes \(e^{y} = x\), making the relationship between the base, power, and result clear and straightforward to work with. This is particularly useful in visualizing growth, as exponential equations depict how quantities change multiplicatively.

• Easier to manipulate algebraically compared to logarithms.
• Useful in real-world applications such as interest calculations and population models.
• Helps confirm the correctness of computations made from logarithmic forms.