Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ \log (5 x+1)=2+\log (2 x-3) $$

Step-by-Step Solution

Answer

The exact solution is \(x = \frac{301}{195}\), approximately \(x \approx 1.54\).

Step 1: Use Logarithmic Properties

Begin by using properties of logarithms. Subtract \(\log (2x-3)\) from both sides of the equation:\[\log (5x+1) - \log (2x-3) = 2\]Use the property \(\log A - \log B = \log \left( \frac{A}{B} \right) \) to combine the logs:\[\log \left( \frac{5x+1}{2x-3} \right) = 2\]

Step 2: Convert Logarithmic to Exponential Form

Convert the logarithmic equation into its exponential form. Recall that if \(\log_{10} (y) = c\), then \(y = 10^c\):\[\frac{5x+1}{2x-3} = 10^2\]Simplify this to:\[\frac{5x+1}{2x-3} = 100\]

Step 3: Solve for \(x\)

Multiply both sides of the equation by \(2x-3\) to clear the fraction:\[5x + 1 = 100(2x - 3)\]Expand and simplify:\[5x + 1 = 200x - 300\]Rearrange the terms to bring like terms together:\[1 + 300 = 200x - 5x\]\[301 = 195x\]Solve for \(x\) by dividing both sides by 195:\[x = \frac{301}{195} \approx 1.54\]

Step 4: Verify Solution

Substitute \(x = 1.54\) back into the original equation to verify the solution. Compute both sides of the original equation using this \(x\) value:- Compute \(\log(5(1.54) + 1)\) and \(\log(2(1.54) - 3)\).- Ensure \(\log(5(1.54) + 1) = 2 + \log(2(1.54) - 3)\).If true, the solution is verified.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make them useful for simplifying complex equations. Understanding these properties is key to efficiently solving logarithmic equations.
One of the primary properties is the subtraction rule, which states:

• If you have \(\log A - \log B\), it can be rewritten as \(\log \left( \frac{A}{B} \right)\).

This property helps us combine logarithms into a single expression. It simplifies the process of solving equations by reducing multiple logarithmic terms into one.
Another essential property is the power rule, though not directly used here, it's good to remember:

• \(\log (A^n) = n * \log (A)\)

These properties allow us to manipulate and solve equations involving logarithms effectively.

Exponential Form

The transition from logarithmic equations to exponential form is a powerful tool for solving these equations. It allows us to remove the logarithm by rewriting the equation using exponents.
The fundamental relation to remember is that if \(\log_{10}(y) = c\), then \(y = 10^c\). This conversion is vital because it transforms a logarithmic equation into an algebraic one.
For instance, in our problem:

• \(\log \left( \frac{5x+1}{2x-3} \right) = 2\) converts to \(\frac{5x+1}{2x-3} = 100\) because \(10^2 = 100\).

This step often simplifies solving the equation since linear equations are generally simpler to handle than logarithmic ones.

Solving Equations

Solving equations, particularly those involving logarithms and exponents, requires systematic steps. Once a logarithmic equation is converted to its exponential form, we can treat it as a standard algebraic equation to find the variable's value.
Here's how we approach this:

• First, clear any fractions by multiplying both sides by the denominator. For example, \(\frac{5x+1}{2x-3} = 100\) becomes \(5x + 1 = 100(2x - 3)\).
• Next, expand and simplify the equation to get all terms involving the unknown on one side, and constants on the other. This yields \(5x + 1 = 200x - 300\).
• Rearrange to isolate the variable \(x\), giving \(301 = 195x\).
• Solve for \(x\) by dividing by the coefficient of \(x\), which results in \(x = \frac{301}{195} \approx 1.54\).

This step-by-step logical approach ensures we get the necessary value, which can then be verified by substituting it back into the initial equation.