Problem 42 - Algebra 2

Question

Solve each equation. $$ \log x-\log 3=8 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3 \times 10^8\)

Step 1: Apply the property of logarithms

To solve the equation, the rule 'the difference of logarithms is the logarithm of the quotient' is applied. Thus, the equation becomes \( \log \frac{x}{3}=8 \).

Step 2: Convert to Exponential Form

Next, change the equation from logarithmic form to exponential form. Hence, \(10^8 = \frac{x}{3}\). This is because the base of the common logarithm is 10.

Step 3: Solve for x

To calculate x, multiply both sides of the equation by 3. This gives x = \(3 \times 10^8\)

Key Concepts

Properties of Logarithms

Logarithms have basic properties that make it easier to simplify and solve logarithmic equations. One crucial property we used in the problem is the "Difference of Logarithms". This states that the difference of two logarithms with the same base can be expressed as the logarithm of a quotient.

• If you see something like \( \log a - \log b \), you can rewrite it as \( \log \left( \frac{a}{b} \right) \).

This simple transformation allows us to combine two logs into one, simplifying the equation before solving. Besides the difference property, there are other useful rules:

• The "Product of Logarithms": \( \log a + \log b = \log(a \cdot b) \)
• The "Power of Logarithms": \( \log(a^b) = b \cdot \log a \)

By using these properties, complex logarithmic equations can often be simplified into a form that's much easier to solve.

Exponential Form Conversion

Converting logarithmic equations into exponential form is like switching from one language to another, allowing us to view the problem in a different way. Remember, logarithms and exponents are two sides of the same coin.

• A logarithmic equation like \( \log_b(x) = y \) can be rewritten in exponential form as \( b^y = x \).

In our exercise, \( \log \left( \frac{x}{3} \right) = 8 \) was rewritten as \( 10^8 = \frac{x}{3} \). Here, '10' is the base of our logarithm (the common logarithm base), and '8' is the exponent to which the base is raised.

• The exponential form provides a clear pathway to isolate and solve for \( x \), turning it from a logarithmic equation to a form that's easier to handle with basic algebra.

Understanding the transformation between logs and exponentials not only simplifies the solving process but also deepens your comprehension of how these two mathematical concepts interrelate.

Solving Equations

Once you've applied the properties of logarithms and converted to an exponential form, you're ready to solve the equation. The process now involves basic algebraic manipulation to find the variable.

• In our problem, after converting \( \log \left( \frac{x}{3} \right) = 8 \) to \( 10^8 = \frac{x}{3} \), solving involves isolating \( x \) on one side of the equation.
• To do this, multiply both sides by 3, the denominator in our fraction. This cancels out the division, leading to \( x = 3 \times 10^8 \).

Solving equations often requires keeping track of inverses and operations to isolate your variable. Multiplying, dividing, adding, and subtracting are basic operations you'll use frequently.

• The goal is to simplify until you have something straightforward, like \( x = ... \).

By methodically applying these steps—property application, conversion, and algebraic manipulation—you can solve even seemingly complex logarithmic equations with confidence.