Problem 90 - Algebra for College Students

Question

Solve each equation. $$ \log _{7} 5+\log _{7} x=1 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{7}{5} \).

Step 1: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \log_b A + \log_b B = \log_b (A \times B) \).Apply this rule to combine the logarithms:\[ \log_7 5 + \log_7 x = \log_7 (5x) \]

Step 2: Equate the Logarithmic Equation with the Given Value

Since \( \log_7 (5x) = 1 \), we can rewrite the equation as:\[ \log_7 (5x) = 1 \]

Step 3: Convert Logarithmic Equation to Exponential Form

The equation \( \log_7 (5x) = 1 \) can be rewritten in exponential form as:\[ 5x = 7^1 \]

Step 4: Solve for x

Since \( 5x = 7^1 \) simplifies to \( 5x = 7 \), divide both sides by 5 to find \( x \):\[ x = \frac{7}{5} \]

Key Concepts

Product Rule of Logarithms

The Product Rule of Logarithms is an important property that helps simplify logarithmic expressions involving sums. If you have two logarithms with the same base added together, like \( \log_b A + \log_b B \), you can combine them into a single logarithm by multiplying the insides: \( \log_b (A \times B) \). This is particularly useful for solving equations as it reduces the complexity of logarithmic expressions.

In the original exercise, we used the Product Rule by taking \( \log_7 5 + \log_7 x \) and turning it into \( \log_7 (5x) \). This consolidation makes it much easier to handle the further steps in solving a logarithmic equation, as it converts multiple terms into a single expression.

Exponential Form

Rewriting a logarithmic equation into its exponential form is another crucial step in solving logarithmic equations. The transformation depends on the concept that if \( \log_b C = D \), then it can be rewritten as \( C = b^D \). This exponential form directly expresses the relationship in terms of multiplication, which is often more straightforward to solve.

For instance, from the step \( \log_7 (5x) = 1 \), we applied the concept of exponential form to change it into \( 5x = 7^1 \), simplifying the equation to a basic arithmetic expression: \( 5x = 7 \). Understanding how to perform this transformation is foundational for steps involving solving for unknown values.

Equation Solving

Solving equations involves isolating the variable in question. Once the logarithmic equation is transformed into its exponential form, you are left with an algebraic equation. The strategy now revolves around basic arithmetic operations.

In our example, after transforming \( \log_7 (5x) = 1 \) to \( 5x = 7 \), the next logical step was simply solving for \( x \) by dividing both sides by 5. This yielded the solution \( x = \frac{7}{5} \).

• Ensure each operation maintains the balance of the equation.
• Perform step-by-step operations: here dividing by 5 on both sides.

Following these steps rigorously helps avoid errors and leads to a smooth solution for the variable \( x \).