Problem 44 - Algebra 2

Question

Solve each equation. $$ 2 \log x+\log 4=2 $$

Step-by-Step Solution

Answer

\[ x = 0.5 \]

Step 1: Combine log expressions

Combine the two logs into one using the property of logarithms: \(\log_a m + \log_a n\) = \(\log_a (m \times n)\). The equation then becomes: \[ \log_2 (2x \times 4) = 2 \] This simplifies to: \[ \log_2 (8x) = 2 \]

Step 2: Convert into an exponential equation

To remove the log, convert the log equation to an exponential equation using the property which states: if \(\log_b (a) = c\), then \(b^c = a\). That gives us:\[ 2^2 = 8x \] Simplify the equation to: \[ 4 = 8x \]

Step 3: Solve for x

Solve the equation for \(x\) by dividing both sides by 8. This gives:\[ x = 4/8 = 0.5 \]

Key Concepts

Properties of Logarithms

Logarithms are fascinating mathematical entities that help us solve complex problems easily. One of the most useful properties of logarithms is the product rule. This rule is fundamental when you need to combine two logarithmic expressions into one.
In the equation from our original exercise, we used this property to merge two logarithmic terms. The product rule states that

• \( \log_a m + \log_a n = \log_a (m \times n) \)

This means if you have two separate logs with the same base, you can combine them.
By applying this rule, we converted \( 2 \log x + \log 4 \) into a simpler form \( \log (8x) \).
This property not only simplifies calculations but also allows us to transform equations into a form that’s easier to work with, like converting a logarithmic equation into an exponential one. Understanding such properties is essential for tackling complex logarithmic expressions with confidence.

Exponential Equations

Exponential equations describe a process where quantities grow or shrink at a constant rate. They are essential in modeling real-world scenarios, such as population growth and radioactive decay.
When we deal with logarithmic equations, transforming them into exponential equations can often simplify the solution process. In our example, we transformed the logarithmic equation \( \log_2 (8x) = 2 \) into an exponential form using the fundamental relationship between logarithms and exponents.

• This relationship states if \( \log_b a = c \), then \( b^c = a \).

By applying this property, we converted the log equation into the exponential form \( 2^2 = 8x \).
This transformation is vital because it simplifies the equation into a straightforward form that eliminates logarithms entirely, paving the way for easier problem-solving.

Solving Equations

Solving equations, whether linear, quadratic, or logarithmic, is all about finding values that satisfy the mathematical statement. In the original exercise involving logarithms, once the equation was rewritten in its simplest form, it was time to solve for \(x\).
We reached an exponential equation \(4 = 8x\), which was much simpler to handle.

• To isolate the variable \(x\), we performed basic algebraic operations.
• This involved dividing both sides of the equation by 8, resulting in \(x = \frac{4}{8} = 0.5\).

Working through these steps methodically simplifies the thinking process.
Remember: Solving equations is about patience and persistence. Always recheck your work to ensure solutions are correct, and take it step-by-step to understand each part of the process deeply.