Problem 26 - College Algebra

Question

For the following exercises, use logarithms to solve. $$ 3^{2 x+1}=7^{x-2} $$

Step-by-Step Solution

Answer

\( x = \frac{-2 \ln(7) - \ln(3)}{2 \ln(3) - \ln(7)} \)

Step 1: Take Logarithm on Both Sides

To solve the equation \( 3^{2x+1} = 7^{x-2} \), take the natural logarithm (or log base 10) on both sides. This gives you: \[ \ln(3^{2x+1}) = \ln(7^{x-2}) \]

Step 2: Apply Logarithmic Identity

Use the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \) to bring down the exponents in front of the logarithms:\[ (2x+1) \cdot \ln(3) = (x-2) \cdot \ln(7) \]

Step 3: Distribute Logarithms

Distribute \( \ln(3) \) and \( \ln(7) \) on both sides:\[ 2x \cdot \ln(3) + 1 \cdot \ln(3) = x \cdot \ln(7) - 2 \cdot \ln(7) \]

Step 4: Collect Like Terms

Rearrange the equation to collect all terms involving \( x \) on one side:\[ 2x \cdot \ln(3) - x \cdot \ln(7) = -2 \cdot \ln(7) - \ln(3) \]

Step 5: Factor Out \( x \) and Solve for \( x \)

Factor \( x \) out of the left side:\[ x(2 \ln(3) - \ln(7)) = -2 \ln(7) - \ln(3) \] Then, solve for \( x \) by dividing both sides by the coefficient of \( x \):\[ x = \frac{-2 \ln(7) - \ln(3)}{2 \ln(3) - \ln(7)} \]

Step 6: Simplify the Expression

Calculate the values for the natural logarithms and simplify the expression if needed to find the numerical approximation of \( x \).

Key Concepts

Exponent Properties

Understanding exponent properties is crucial in solving many mathematical problems, especially when dealing with exponential and logarithmic equations. Exponents represent repeated multiplication. For example, an expression like \( a^b \) means multiplying \( a \) by itself \( b \) times. Here are some key properties of exponents that help in simplifying expressions:

• Product of Powers: \( a^m \times a^n = a^{m+n} \) - when you multiply like bases, you add the exponents.


• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \) - when you divide like bases, you subtract the exponents.


• Power of a Power: \( (a^m)^n = a^{m \cdot n} \) - when raising a power to another power, you multiply the exponents.


• Power of a Product: \( (ab)^n = a^n b^n \) - when you raise a product to a power, you apply the exponent to each factor.


• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \) - an exponent can turn a number into its reciprocal if the exponent is negative.

These properties are instrumental in manipulating and simplifying expressions, as seen in the equation \( 3^{2x+1} = 7^{x-2} \). Raising each side to the power simplifies solving through natural logarithms.

Natural Logarithm

Natural logarithms are logarithms with a base of \( e \), where \( e \) is approximately 2.718. The natural logarithm is denoted by \( \ln \). When solving equations, like \( 3^{2x+1} = 7^{x-2} \), natural logarithms help in handling the exponential expressions.

• Comprehension: The natural logarithm \( \ln(x) \) asks, "To what power must \( e \) be raised to yield \( x \)?"


• Popularity: Natural logarithms are highly prevalent in calculus and mathematical modeling due to their relation with exponential growth and decay.


• Inverse Function: The natural log function \( \ln(x) \) is the inverse of the exponential function \( e^x \), meaning \( \ln(e^x) = x \) and \( e^{\ln x} = x \) for \( x > 0 \).

Using logarithmic identities, like \( \ln(a^b) = b \cdot \ln(a) \), the natural log allows isolating the variable \( x \), simplifying complex exponential equations to solve for \( x \).

Logarithmic Identities

Logarithmic identities are versatile tools for transforming logarithmic expressions, particularly useful in solving equations. By converting multiplication into addition, division into subtraction, and handling exponents deftly, these identities unravel complex mathematical scenarios.When we tackled the equation \( 3^{2x+1} = 7^{x-2} \), the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \) played a central role. This transforms the equation by pulling down exponents, a crucial step for solving it. Here are some common logarithmic identities:

• Power Rule: \( \log_b(a^c) = c \cdot \log_b(a) \)


• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)


• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)


• Base Change Rule: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \)

Applying these identities allows for the decomposition of logarithmic equations into more manageable parts. By understanding and using these identities effectively, you can solve logarithmic equations with ease.