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When working with logarithmic equations, understanding the properties of logarithms is crucial. Logarithms are mathematical entities that are essentially the inverse of exponentiation. This means if you have a number expressed as a power, a logarithm can help you find the original number.

Key logarithmic properties include:

• Product Property: This property states that the logarithm of a product equals the sum of the logarithms of the factors, written as: \( \log(a \times b) = \log a + \log b \).
• Quotient Property: This one states that the logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator, given by: \( \log(\frac{a}{b}) = \log a - \log b \).
• Power Property: This property shows that the logarithm of a power is the exponent times the logarithm of the base, represented as: \( \log(a^b) = b \cdot \log a \).

Understanding these properties is essential for simplifying and solving logarithmic equations effectively.

The product rule of logarithms is one of the core concepts when dealing with logarithmic equations. This rule helps in rewriting the sum of two logarithms into a single logarithmic expression. It states: \( \log a + \log b = \log (a \times b) \). This rule allows us to combine two separate logarithmic terms into one, making it easier to solve equations.

Let's look at how this rule impacted our exercise:

• The mistake was in combining \( \log x + \log 2 \) incorrectly into \( \log(x+2) \), treating it as a single logarithm with addition inside. That's incorrect.
• The correct approach is applying the product rule to rewrite it as \( \log (x \times 2) \), which simplifies to \( \log 2x \).
• This simplification allows the equation \( \log 2x = \log 5 \) to be effectively solved using another property.

Understanding and correctly applying the product rule can prevent common mistakes in solving logarithmic equations.

Once you've simplified a logarithmic equation using the product rule or other properties, the next step is to solve the equation. Solving logarithmic equations often involves using the property of one-to-one functions of logarithms.

The one-to-one property indicates that if two logarithms with the same base are equal, then the arguments must be equal. In simple terms, \( \log a = \log b \) implies that \( a = b \). This is essential for moving from a logarithmic equation to a simpler algebraic equation.

In the refined solution from our exercise, we end up with \( \log(2x) = \log(5) \). By applying the one-to-one property, we determine that \( 2x = 5 \).

Finally, you solve this algebraic equation by isolating \( x \). Divide both sides by 2 to obtain \( x = \frac{5}{2} \), or equivalently, \( x = 2.5 \).

This simplification and solving process highlights the importance of understanding both logarithmic and algebraic properties for correctly solving logarithmic equations.