Problem 89 - Algebra for College Students

Question

Solve each equation. $$ \log _{3} x+\log _{3} 4=2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{9}{4} \).

Step 1: Use the Property of Logarithms

We will use the property of logarithms which states that the sum of the logs is the log of the product: \( \log_b a + \log_b c = \log_b(ac) \). In this equation, we can combine the logarithms on the left side: \( \log_{3} x + \log_{3} 4 = \log_{3}(4x) \).

Step 2: Simplify the Equation

Now that the left side of the equation is simplified, the equation becomes: \( \log_{3}(4x) = 2 \).

Step 3: Rewrite the Equation in Exponential Form

Use the inverse property of logarithms to rewrite the equation in exponential form: If \( \log_{b} a = c \), then \( a = b^c \). Apply this to \( \log_{3}(4x) = 2 \), resulting in \( 4x = 3^2 \).

Step 4: Solve for x

Calculate \( 3^2 \) to get 9, so the equation becomes: \( 4x = 9 \). Divide both sides by 4 to solve for \( x \): \( x = \frac{9}{4} \).

Step 5: Verify the Solution

Substitute \( x = \frac{9}{4} \) back into the original equation to check if it holds true: \( \log_{3}\left(\frac{9}{4}\right) + \log_{3}(4) = \log_{3}(9) \). Since \( 3^2 = 9 \), \( \log_{3}(9) = 2 \). Both sides of the equation are equal, confirming that \( x = \frac{9}{4} \) is correct.

Key Concepts

Properties of Logarithms

Logarithms have special properties that make them useful for solving equations. One key property is the product property, which states:

• \( \log_b a + \log_b c = \log_b(ac) \)

This means that if you are adding two logarithms with the same base, you can combine them into a single logarithm by multiplying the two arguments together.
In the original exercise, this property is used to combine \( \log_{3} x \) and \( \log_{3} 4 \). By applying this property, the two terms are combined into \( \log_{3}(4x) \).
This simplification is crucial because it allows us to handle the problem as a single logarithmic expression, simplifying our task of finding the solution.

Exponential Form

Once you have an equation in the form \( \log_{b} a = c \), you can rewrite it in exponential form. This uses the idea that logarithms and exponents are inverse functions. The rule to remember is:

• If \( \log_b a = c \), then \( a = b^c \).

Converting from a logarithmic form to an exponential form allows us to solve the equation for unknown variables.
In the example given, thanks to applying the product property, we rewrite \( \log_{3}(4x) = 2 \) as \( 4x = 3^2 \).
Being able to switch between these forms is a powerful tool, especially when solving equations that involve logarithms.

Solving Equations

With an equation expressed in exponential form, you can solve for the unknown variable through simple algebraic manipulation.

• Calculate any powers, like in our example, where \( 3^2 \) gives us 9.
• Then, isolate the variable by performing operations such as division or multiplication.

For the exercise, after rewriting in exponential form, you get \( 4x = 9 \). Divide both sides by 4 to isolate \( x \), resulting in \( x = \frac{9}{4} \).
Finally, confirm your solution is correct by substituting it back into the original equation. Ensuring the original equation holds verifies that the computed solution is accurate.