Problem 23 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \ln x=\ln 5+\ln 9 $$

Step-by-Step Solution

Answer

The solution is \( x = 45 \).

Step 1: Understand the given equation

The given logarithmic equation is \( \ln x = \ln 5 + \ln 9 \). We need to find the value of \( x \). The equation on the right-hand side involves the sum of two logarithms.

Step 2: Apply properties of logarithms

Use the property of logarithms that states \( \ln a + \ln b = \ln(ab) \). Applying this property, the equation becomes \( \ln x = \ln(5 \times 9) \).

Step 3: Simplify the equation

Simplify the expression on the right-hand side to get \( \ln x = \ln 45 \). This equation now equates the logarithms of two numbers.

Step 4: Solve for x

Since the natural logarithm function, \(\ln\), is a one-to-one function, we can deduce that \( x = 45 \). We equated the expressions inside the logarithms, leading us directly to the solution.

Key Concepts

Properties of Logarithms

Logarithmic properties are essential tools in simplifying logarithmic expressions and solving equations. One of the most useful properties is the product rule. This rule states that the logarithm of a product is the sum of the logarithms of its factors:
\[\ln(ab) = \ln a + \ln b\]
This property enables you to combine two separate logarithms into one, simplifying expressions where possible.

• The property works for any bases of logarithms, as long as they are the same.
• Understanding this property can help you solve more complex logarithmic equations by reducing the number of terms.

Another important property is the quotient rule, similar to the product rule:
\[\ln\left(\frac{a}{b}\right) = \ln a - \ln b\]
There is also the power rule:
\[\ln(a^b) = b\ln a\]
By mastering these basic properties, solving logarithmic equations becomes a streamlined process, as they allow for manipulation of logarithmic expressions into simpler forms.

Solving Logarithmic Equations

To solve a logarithmic equation, the first step is to simplify the equation using the properties of logarithms. In our example, \( \ln x = \ln 5 + \ln 9 \), we use the product rule to combine the logarithms on the right side:
\[ \ln x = \ln(5 \times 9) \]
Which simplifies to:
\[ \ln x = \ln 45 \]
With the logarithms on both sides simplified to the same form, we can compare the expressions inside the logarithms.

• Since the logarithm function is one-to-one, if \( \ln a = \ln b \), then \( a = b \).
• This allows us to conclude that \( x = 45 \), directly solving our equation.

Approaching logarithmic equations step-by-step ensures accuracy. Start by identifying applicable properties to simplify both sides. Then evaluate when functions can be equal under the given conditions.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific type of logarithm with a special base known as Euler's number \( e \), approximately 2.718.
It's called "natural" because it naturally occurs in many mathematical contexts, like compound interest formulas and calculus.

• Natural logarithms are used to solve equations involving exponential growth and decay.
• They provide a straightforward way to reverse the effects of exponential functions.

In equations, the natural logarithm has unique properties that starkly mirror those of regular logarithms:

• \( \ln e = 1 \), meaning the natural log of its base, \( e \), equals 1.
• It helps when transforming multiplicative relationships into additive ones.

This makes them very practical for solving various types of mathematical problems, making \( \ln \) a powerful tool in both pure and applied mathematics.