Problem 35 Solve each equation. See Example... [FREE SOLUTION]

Chapter 9: Problem 35

Solve each equation. See Examples 1 through 4. $$ \log _{4} x-\log _{4}(2 x-3)=3 $$

Short Answer

Expert verified

The solution is $x = \frac{192}{127}$.

Step by step solution

Apply Logarithmic Identity

Recall the property of logarithms: $ \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) $. We can use this identity to combine the two logarithms on the left hand side of the equation: \[ \log_{4} x - \log_{4} (2x - 3) = \log_{4} \left( \frac{x}{2x - 3} \right) \] Thus, the equation becomes: \[ \log_{4} \left( \frac{x}{2x - 3} \right) = 3 \]

Convert to Exponential Form

To solve the equation $ \log_{4} \left( \frac{x}{2x - 3} \right) = 3 $, convert the logarithmic equation to its exponential form. The exponential form of $ \log_{b} a = c $ is $ a = b^c $. \[ \frac{x}{2x - 3} = 4^3\] Calculate $4^3$: \[ 4^3 = 64 \] So we have: \[ \frac{x}{2x - 3} = 64 \]

Solve for x

Now solve the equation $ \frac{x}{2x - 3} = 64 $. Start by multiplying both sides of the equation by $(2x - 3)$ to clear the fraction:\[ x = 64(2x - 3) \] Expand the right side:\[ x = 128x - 192 \] Rearrange terms to isolate $x$:\[ x - 128x = -192 \] Combine like terms:\[ -127x = -192 \] Divide by -127:\[ x = \frac{-192}{-127} = \frac{192}{127} \]

Verify the Solution

Substitute $x = \frac{192}{127} $ back into the original logarithmic equation: \[ \log_{4} \frac{192}{127} - \log_{4}(2 \times \frac{192}{127} - 3) = 3 \] Calculate:\[ 2 \times \frac{192}{127} - 3 = \frac{384}{127} - \frac{381}{127} = \frac{3}{127} \] So: \[ \log_{4} \frac{192}{127} - \log_{4} \frac{3}{127} = \log_{4} \frac{64}{1} = \log_{4} 64 \] Since $\log_{4} 64 = 3$, the left side equals the right side, verifying the solution. Thus, $x = \frac{192}{127}$ is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Identities

In mathematics, understanding the properties of logarithms can save you a lot of time and effort when solving equations. A key property to remember is the logarithm subtraction identity: $ \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) $. This identity allows us to combine two logarithms with the same base into a single logarithm, simplifying the expression.

It helps in reducing the complexity of equations where logarithms appear on both sides.
Once combined, it is often easier to manipulate and solve the equation as a whole.

In our exercise, when faced with the equation $ \log_{4} x - \log_{4} (2x - 3) = 3 $, we apply this identity to transform the left side into $ \log_4 \left( \frac{x}{2x - 3} \right) $. This consolidation paves the way for further steps in solving the equation.

Solving Logarithmic Equations

With the logarithmic identities simplified, the next step is solving the equation. A logarithmic equation typically requires converting it into a more manageable form to find the solution.

Identify the logarithmic form and recognize applicable identities or properties.
Convert it into an exponential form, which is often easier to solve.

In our example, the equation becomes $ \log_4 \left( \frac{x}{2x - 3} \right) = 3 $. At this point, the equation is ready to be solved using other algebraic techniques after conversion, such as solving linear equations or applying cross-multiplication to clear fractions.

Exponential Form

Converting a logarithmic equation into its exponential form is a critical step in solving the equation. This conversion directly uses the definition: $ \log_b a = c $ is equivalent to $ a = b^c $.

This step simplifies the arithmetic involved by transforming a logarithm problem into an exponent problem.
Exponential equations are typically solved by isolating the variable or using techniques like factorization.

For the equation $ \log_4 \left( \frac{x}{2x - 3} \right) = 3 $, we convert it to $ \frac{x}{2x - 3} = 4^3 $, which simplifies to $ \frac{x}{2x - 3} = 64 $. This form allows us to use algebraic methods like multiplying both sides to clear fractions and solve for $ x $.

Verifying Solutions

Finding a solution is essential, but verification ensures that the solution you've found is correct. Verifying requires substituting the solution back into the original equation.

This process helps confirm that the solution satisfies the initial condition set by the equation.
It also checks for extraneous solutions that might not have been apparent initially.

To verify $ x = \frac{192}{127} $ in our exercise, substitute back into the original logarithmic equation: $ \log_4 \frac{192}{127} - \log_4 \left(2 \times \frac{192}{127} - 3\right) = 3 $. After simplification, it equals $ \log_4 64 $, which is indeed 3. This confirms the solution is correct and satisfies the equation as intended.

91影视

Solve each equation. See Examples 1 through 4. $$ \log _{4} x-\log _{4}(2 x-3)=3 $$

Short Answer

Step by step solution

Apply Logarithmic Identity

Convert to Exponential Form

Solve for x

Verify the Solution

Key Concepts

Logarithmic Identities

Solving Logarithmic Equations

Exponential Form

Verifying Solutions

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