Question

\(\log 0.786\) is equal to : (a) \(\log 786-\log 3\) (b) \(\log 78.6-\log 100\) (c) \(\log 786-3\) (d) both (b) and (c)

Short answer

a) \(\log 786 - \log 3\) b) \(\log 78.6 - \log 100\) c) \(\log 786 - 3\) d) Both (b) and (c) Answer: b) \(\log 78.6 - \log 100\)

Step-by-step solution

Identifying the properties of logarithms

The properties of logarithms that we will use for this problem are: 1. \(\log(ab) = \log a + \log b\) 2. \(\log\frac{a}{b} = \log a - \log b\) 3. \(\log a^b = b\log a\)

Compare each option with the original expression

We will now compare each option with \(\log 0.786\) by simplifying them using the properties of the logarithm. For option (a): \(\log 786 - \log 3\) Using property 2: \(\log\frac{786}{3} = \log 262\) Since \(\log 262\) doesn't equal \(\log 0.786\), option (a) is incorrect. For option (b): \(\log 78.6 - \log 100\) Using property 2: \(\log\frac{78.6}{100} = \log 0.786\) Since \(\log 0.786\) equals the original expression, option (b) is correct. For option (c): \(\log 786 - 3\) Using property 3: \(\log 786^{\frac{1}{3}} = \log 9.235...\) Since \(\log 9.235...\) doesn't equal \(\log 0.786\), option (c) is incorrect. For option (d): Both (b) and (c) Since we determined that option (c) is incorrect, option (d) is also incorrect.

Choose the correct option

Based on our comparisons, the correct equivalency for \(\log 0.786\) is option (b): \(\log 78.6 - \log 100\).

Key concepts

Logarithm Properties

Understanding the properties of logarithms is crucial when we want to simplify or equate logarithmic expressions. Here are some key properties you need to know:

• Product Property: If you have two numbers, say \(a\) and \(b\), multiplied inside a logarithm, you can split them up as \(\log(ab) = \log a + \log b\).
• Quotient Property: If you have a fraction inside a logarithm, you can express it as \(\log\frac{a}{b} = \log a - \log b\). This property helps break down division within logarithmic problems.
• Power Property: When dealing with exponents, you can bring the exponent in front of the logarithm: \(\log a^b = b \log a\).

These properties are powerful tools that allow you to manipulate logarithmic expressions, making it easier to compare, simplify, and solve logarithmic equations.

Equivalent Expressions

To determine if two expressions are equivalent, we must simplify them to see if they result in the same value. This is especially useful in logarithmic equations where properties can transform one expression into another. For instance, if you're given a logarithmic expression and asked to find an equivalent form, you should:

• Use the Quotient Property to split or combine fractions in logarithms.
• Apply the Product Property when dealing with multiplication inside a logarithm.
• Leverage the Power Property for expressions involving exponents.

By applying these properties, you can compare the given options in a problem to the original expression—like in our exercise. It allows you to identify which options hold true as equivalent expressions, leading you to the correct solution.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms of variables. Solving these equations often requires the use of logarithm properties to simplify and manipulate them into a form that is easier to solve. Typically, the process involves:

• Simplification: Use logarithm properties to combine or separate logarithmic terms.
• Conversion: Sometimes, converting a logarithmic equation into an exponential form can simplify the solving process.
• Isolation: Try to isolate the logarithm on one side to make it easier to solve for the variable.

Throughout this process, having a strong understanding of logarithm properties is essential. It allows for moving seamlessly between different forms of the equation until it becomes solvable. Therefore, mastering these skills is fundamental for effectively solving logarithmic equations.