Problem 12 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 6^{x+3}=2 $$

Step-by-Step Solution

Answer

Exact: \(x = \frac{\ln(2)}{\ln(6)} - 3\); Approximate: \(-2.6131\).

Step 1: Set up the equation

The given equation is \(6^{x+3} = 2\). We want to solve for \(x\).

Step 2: Apply the logarithm to both sides

Take the natural logarithm of both sides to bring down the exponent: \(\ln(6^{x+3}) = \ln(2)\).

Step 3: Use logarithm power rule

By the power rule of logarithms, \(\ln(6^{x+3}) = (x+3)\ln(6)\). Therefore, the equation becomes \((x+3)\ln(6) = \ln(2)\).

Step 4: Solve for \(x\)

Isolate \(x\) by first dividing both sides by \(\ln(6)\): \(x+3 = \frac{\ln(2)}{\ln(6)}\). Then, subtract 3 from both sides: \(x = \frac{\ln(2)}{\ln(6)} - 3\).

Step 5: Calculate the exact solution

The exact expression for \(x\) is \(x = \frac{\ln(2)}{\ln(6)} - 3\).

Step 6: Calculate the approximate solution

Using a calculator, find the approximate value. First compute \(\ln(2) \approx 0.6931\) and \(\ln(6) \approx 1.7918\). Then substitute to find \(x \approx \frac{0.6931}{1.7918} - 3 \approx -2.6131\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse operations of exponentiation, and they help us solve equations where the variable is an exponent. In practice, a logarithm answers the question: "To what exponent must we raise a specific base to obtain a certain number?" If we have an equation like \( b^x = y \), the logarithm with base \( b \) is expressed as \( x = \log_b(y) \). This is quite valuable when working to solve exponential equations. For example, by taking the logarithm of both sides of an equation like \( 6^{x+3} = 2 \), we transform it into a linear form that is easier to solve.

• Logarithm bases: Common bases used in logarithms are 10 (common logarithm), the number e (natural logarithm), and 2, among others.
• Properties: The power rule, \( \log_b(m^n) = n\cdot\log_b(m) \), lets us bring down exponents as coefficients, simplifying equations for easier calculation.
• Applications: Logarithms are used in calculating exponential decay, population growth, and solving exponential equations in many real-world scenarios.

Learning how to manipulate and interpret logarithmic functions gives you the power to solve complex equations that involve exponentiation, making them a crucial tool in mathematics.

Exponentiation

Exponentiation is the mathematical operation that involves raising one number, known as the base, to the power of another number, which is the exponent. It is a fundamental concept often encountered in various areas of mathematics and has practical applications in science and engineering. When we write \( b^x \), "b" is the base, and "x" is the exponent.

• Multiplying powers: The base raised to an exponent is a multiplication of that base done as many times as indicated by the exponent, i.e., \( b^n = b \times b \times ... \times b \) (n times).
• Properties: Some key properties include \( b^0 = 1 \) for any non-zero base "b", and \(b^m \cdot b^n = b^{m+n} \).
• Exponential growth and decay: In real-world applications, exponentiation models scenarios like population growth (growth factor) and radioactive decay (decay factor), showcasing how quantities grow or shrink exponentially.

Understanding exponentiation is essential for working with exponential equations, as it helps form the pathway to explore exponential growth and solutions more thoroughly.

Natural Logarithm

The natural logarithm, often denoted as \( \ln(x) \), is a special logarithm with the base of "e", where "e" is an irrational mathematical constant approximately equal to 2.71828. Unlike other logarithms where base 10 might be used, the natural logarithm is particularly significant in calculus and complex equations.

• Definition: If \( x = e^y \), then \( y = \ln(x) \). Here, the natural logarithm gives us the power to which "e" must be raised to obtain the number "x".
• Common uses: Natural logs simplify the derivatives and integrals of exponential functions, making them indispensable in the study of calculus.
• Change of base formula: Allows conversion between different logarithm bases: \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \), which can simplify calculations when solving equations.

In mathematical contexts, natural logarithms provide a simpler framework for analysis, especially when dealing with growth rates, decay, or solving for unknown variables in exponential relationships.