Problem 46 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$1-4 \ln (2 x-1)=-5$$

Step-by-Step Solution

Answer

\(x = \frac{e^{3/2} + 1}{2}\)

Step 1: Isolate the Logarithmic Term

To solve the logarithmic equation, we first need to isolate the logarithmic term. Starting from the equation: \[1 - 4 \ln(2x - 1) = -5\]Subtract 1 from both sides:\[-4 \ln(2x - 1) = -6\]

Step 2: Divide Both Sides by -4

Now, divide both sides of the equation by -4 to solve for the natural logarithm:\[\ln(2x - 1) = \frac{6}{4} = \frac{3}{2}\]

Step 3: Exponentiate to Eliminate the Logarithm

To eliminate the logarithm, exponentiate both sides using the base of the natural logarithm, which is \(e\). Hence, \[e^{\ln(2x-1)} = e^{\frac{3}{2}}\]This simplifies to:\[2x - 1 = e^{\frac{3}{2}}\]

Step 4: Solve for x

Now solve for \(x\) by isolating it on one side of the equation:First, add 1 to both sides:\[2x = e^{\frac{3}{2}} + 1\]Then divide by 2:\[x = \frac{e^{\frac{3}{2}} + 1}{2}\]

Step 5: Verify the Solution with a Calculator

Use a calculator to compute \(e^{\frac{3}{2}}\). If necessary, convert the result to a numerical value to verify the final solution. This checks that the solution satisfies the original equation as closely as possible, taking into account any rounding errors.

Key Concepts

Natural Logarithm

Natural logarithms are logs to the base of e, where e is approximately equal to 2.71828. The symbol for natural logarithm is \( \ln \). When you see \( \ln(x) \), it means "the power to which we must raise the number e to get x." Natural logarithms are particularly useful in calculus and can simplify complex multiplicative processes with their additive properties.

• Given \( \ln(a) = b \), it indicates that \( e^b = a \).
• In the logarithmic equation \( \ln(2x - 1) = \frac{3}{2} \), it tells us that the expression \( 2x - 1 \) can be rewritten in terms of the exponential function, namely: \( e^{\frac{3}{2}} \).

Understanding natural logarithms is crucial as they appear frequently in real-world situations involving exponential growth and decay, such as in population growth and radioactive decay.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself. For example, in \( 2^3 \), 2 is the base, and 3 is the exponent, meaning \( 2 \times 2 \times 2 \).

• The property \( e^{\ln(x)} = x \) allows us to "undo" the logarithm. That's why in our step-by-step solution, we use exponentiation to solve \( \ln(2x - 1) = \frac{3}{2} \). Both sides of the equation are raised to the power of e.
• Applying exponentiation helps remove the logarithm, simplifying the expression to \( 2x - 1 = e^{\frac{3}{2}} \).

Remember, exponentiation is the inverse operation of the logarithm, and this relationship is key to solving logarithmic equations.

Exact Form Solutions

In mathematics, an exact form solution provides the answer in terms of constants and variables without numerical approximations. Instead of converting \( e^{\frac{3}{2}} \) into a decimal, exact form means leaving it as \( e^{\frac{3}{2}} + 1 \). It provides exact values that are often more insightful, as any numerical approximation could introduce error.

• In the logarithmic equation solution, we find \( x = \frac{e^{\frac{3}{2}} + 1}{2} \).
• This expression is exact, relying on the fundamental constants like e, which is crucial for maintaining precise information in scientific and mathematical work.

Always aim for exact form solutions when possible, unless a numerical approximation is necessary for a practical decision or computation.

Use of Calculator in Algebra

Using a calculator in algebra can be invaluable when verifying results or converting exact expressions into decimal form. For example, when we solve for \( x \) in the equation and find \( x = \frac{e^{\frac{3}{2}} + 1}{2} \), using a calculator allows us to compute \( e^{\frac{3}{2}} \) to get a numerical approximation for practical interpretation.

• Enter expressions carefully into the calculator to avoid errors. Ensure you're using parentheses correctly to maintain mathematical accuracy.
• Use calculators to check your algebra by comparing the approximated decimal values against your expectations from the exact form solution.

Calculators are tools to aid in understanding, not replace fundamental algebraic expertise. Knowing when and how to use them enhances problem-solving efficiency.