Problem 64 - Algebra 2

Question

Solve each equation. $$ \ln x-\ln 4=5 $$

Step-by-Step Solution

Answer

The value of x that satisfies the equation is \( x = 4e^5 \).

Step 1: Spot Logarithmic Identity

Observe that the equation contains two logarithms that are subtracted. Remember that the subtraction of two logarithms can be rewritten using the following rule: \( \ln a - \ln b = \ln (\frac {a}{b}) \). Utilize this rule to rewrite the equation. The equation becomes: \( \ln \frac {x}{4} = 5 \).

Step 2: Remove the Logarithm

The goal is to solve for x, so the logarithm needs to be removed to carry out further operations. Apply the anti-logarithm to both sides of the equation, which simply means rewrite the equation without the logarithm. Since the base of a natural logarithm (ln) is \( e \), the anti-logarithm of \( \ln \frac {x}{4} = 5 \) is \( \frac {x}{4} = e^5 \).

Step 3: Solve for x

Now that \( \frac {x}{4} = e^5 \) is the equation, do some algebra to get the x alone. Multiply both sides of the equation by 4, yielding the final solution: \( x = 4e^5 \)

Key Concepts

Logarithmic Identities

When solving logarithmic equations, logarithmic identities are incredibly handy. These identities make it easier to manipulate and simplify logarithmic expressions. One of the most common identities is the difference of logarithms, which states that \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \).
This identity helps condense the subtraction of two logs into a single log. Understanding this concept simplifies solving equations involving multiple logs.

• Make use of logarithmic identities to reduce complexity.
• Ensure your logarithms have the same base when applying identities.
• Recognize when a logarithmic identity applies to avoid unnecessary complications.

By leveraging identities, you'll often find that problems become much easier and straightforward.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific type of logarithm. Its base is the irrational number \( e \), which is approximately equal to 2.718.
Natural logarithms are commonly used in mathematics due to their natural properties that simplify many equations and mathematical models.

• \( \ln e = 1 \) because the base \( e \) raised to the power of 1 gives \( e \).
• Natural logs are the inverse operation of the exponential function with base \( e \).
• They are frequently used alongside exponential functions to solve equations, especially in calculus and higher-level mathematics.

Understanding how to manipulate and simplify equations involving natural logs is essential for various applications in science and engineering.

Exponential Functions

Exponential functions are vital in mathematics and describe situations where a quantity grows or decays at a rate proportional to its current value. A common form of an exponential function is \( b^x \), where \( b \) is the base.
The base for natural exponential functions is the number \( e \). These functions are particularly famous for modeling continuous growth or decay.

• An exponential function with base \( e \) is expressed as \( e^x \).
• They are closely linked to their inverses, the logarithmic functions.
• Exponential growth follows a pattern where quantities increase steadily and predictably.

Exponential functions are crucial because they model real-world phenomena, such as population growth, radioactive decay, and interest calculations, making them an integral part of scientific and economic studies.