Problem 61 - Algebra 2

Question

Solve each equation. $$ \frac{1}{3} \ln x+\ln 2-\ln 3=3 $$

Step-by-Step Solution

Answer

The solution is \(x = e^{9} * (27/8)\)

Step 1: Simplify the equation

Use the properties of logarithms to simplify the given equation. For this case, we combine the \( \ln \) terms using the additional logarithm property where \( \ln ab = \ln a + \ln b \). Hence, we get \(\frac{1}{3}\ln x + \ln(2/3) = 3 \)

Step 2: Move all the logarithmic expressions to one side

Next, isolate logarithmic expressions on the one side of equation. This results to: \( \frac{1}{3}\ln x = 3 - \ln(2/3) \)

Step 3: Solve for the variable

Multiply both sides by 3 to get \( \ln x = 3(3 - \ln(2/3)) \)

Step 4: Convert the logarithmic equation to its equivalent exponential form

Continue with changing the base the formula from logarithm to its equivalent exponential form to solve for x using the formula \(b^{ \ln_b(a) } = a\). Thus, \(x = e^{9 - 3\ln(2/3)}\)

Step 5: Simplificate the answer

Applying the power rule for exponential functions to simplificate the expression: \(x = \frac{e^{9}}{(e^{\ln(2/3)})^3}\). Further simplification results in: \(x = \frac{e^{9}}{(2/3)^3}\). Fully simplified: \(x = e^{9} * (27/8)\)

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is crucial when solving equations involving logarithms. The properties allow us to simplify expressions and solve logarithmic equations more efficiently. Here are some important logarithmic properties:

• Product Property: This property states that \(\ln(a \, b) = \ln a + \ln b\)\. It lets us combine the logarithms of multiplied numbers.


• Quotient Property: This states \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\)\. It allows us to deal with the logarithm of a quotient.


• Power Property: According to this, \(\ln(a^b) = b \cdot \ln a\)\. It helps when a logarithmic term has an exponent.

Applying these properties makes it easier to manipulate and solve logarithmic expressions by expressing them in simpler forms.

Exponential Form

Transforming logarithmic equations to exponential form is an effective way to solve for the variable, especially when it is within a logarithmic function. A logarithm is essentially the opposite, or inverse, of an exponential function. The core principle is that if \(\ln_b(a) = c\), then \(b^c = a\).
This relationship allows us to switch from complex logarithmic expressions to simpler exponential ones:

• Converting to Exponential Form: For example, if you have \(\ln x = 3\), then in exponential form, it becomes \(e^3 = x\), since 'e' is the base of the natural logarithm.


• This transformation is particularly useful when calculations involve solving for the unknown, as exponential forms are often easier to evaluate using basic calculations.

Employing exponential form strategically leads to immediate simplification in the problem-solving process.

Exponential Functions

Exponential functions are mathematical expressions where variables appear as exponents. They have a constant base raised to a variable power, such as \(y = a^x\). These functions grow or decay exponentially, showcasing rapid increases or decreases.
Understanding exponential functions is fundamental because:

• They provide models for many real-world phenomena, like population growth or radioactive decay.


• They are used in various mathematical contexts, such as transforming logarithmic equations into more manageable forms.


• In terms of solving equations, understanding the nature of these functions aids in recognizing the relationship between the base and the exponent, which is vital when interpreting results or seeking solutions.

Their predictable patterns make them key tools in both academic exercises and practical applications.