Problem 64 Explain why $$ \frac{1+\log ... [FREE SOLUTION]

Chapter 3: Problem 64

Explain why $$ \frac{1+\log x}{2}=\log \sqrt{10 x} $$ for every positive number $x$

Short Answer

Expert verified

We can show that the given equation $\frac{1+\log{x}}{2} = \log{\sqrt{10x}}$ holds for every positive number $x$ through simplification and the use of logarithmic properties: 1. Rewrite the right side of the equation as $\frac{1}{2}\log{(10x)}$. 2. Distribute the fraction on the right side to obtain $\frac{1}{2}\log{10} + \frac{1}{2}\log{x}$. 3. Recall that $\log{10} = 1$, simplifying the equation to $\frac{1+\log{x}}{2} = \frac{1}{2} + \frac{1}{2}\log{x}$. By comparing the two expressions, we can see that they are equivalent, proving that the equation holds for every positive number $x$.

Step by step solution

Rewrite the given equation

Let's first rewrite the equation and use logarithm properties: $$ \frac{1+\log{x}}{2} = \log{\sqrt{10x}} $$

Simplify the square root using logarithmic properties

Use the logarithm property $\log{a^b} = b\log{a}$ to simplify the right side of the equation: $$ \frac{1+\log{x}}{2} = \frac{1}{2}\log{(10x)} $$

Distribute the fraction

Distribute the fraction on the right side of the equation: $$ \frac{1+\log{x}}{2} = \frac{1}{2}\log{10} + \frac{1}{2}\log{x} $$

Simplify the logarithmic expressions

Remember that $\log{10} = 1$, hence the equation simplifies to: $$ \frac{1+\log{x}}{2} = \frac{1}{2} + \frac{1}{2}\log{x} $$

Compare the expressions

Now we can see that the two expressions are the same: $$ \frac{1+\log{x}}{2} = \frac{1}{2} + \frac{1}{2}\log{x} $$ We have proven that the equation holds for every positive number $x$, as both expressions are equivalent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Properties

Logarithmic properties are extremely useful in simplifying and solving logarithmic equations. These properties allow us to manipulate logarithmic expressions into different forms that are easier to work with. Here are some key properties that you should always keep in mind:

**Log of a Power**: $ \log{a^b} = b\log{a} $ - This property is especially useful when dealing with expressions involving roots or powers, as in this exercise where $ \sqrt{10x} $ can be rewritten using this property.

**Product Property**: $ \log{(ab)} = \log{a} + \log{b} $ - This is handy when simplifying products inside a logarithm.

**Quotient Property**: $ \log{\left(\frac{a}{b}\right)} = \log{a} - \log{b} $ - Use this property to simplify quotients.

**Change of Base Formula**: Although not used directly in this example, knowing $ \log_b{a} = \frac{\log_c{a}}{\log_c{b}} $ can be useful when dealing with logs of different bases.

These properties make it possible to rewrite complex logarithmic expressions into simpler forms, enabling easier comparison or solution of the equations.

Equations

Equations involving logarithms can initially seem daunting, but with a strategic approach, they become manageable. Solving logarithmic equations often requires manipulating expressions using logarithmic properties and algebraic techniques until you can clearly see the solution.

In the exercise, the equation $ \frac{1 + \log x}{2} = \log \sqrt{10 x} $ was solved by first rewriting and simplifying both sides using logarithmic properties.

The right side of the equation was simplified by employing the property $ \log a^b = b \log a $, converting $ \log \sqrt{10x} $ into $ \frac{1}{2} \log(10x) $.

At every step, both sides of the equation were made as similar as possible, allowing for a direct comparison and verification of equality.

Through these techniques, we can prove the equality of both sides of a logarithmic equation for all valid values of $ x $, as seen in the successful solution of this exercise.

Logarithmic Expressions

Logarithmic expressions like $ 1 + \log x $ can sometimes appear complex, but they are expressions that incorporate logarithms and can often be broken down into simpler components. Understanding and manipulating these expressions is key to solving many exponential and logarithmic problems.

In the given exercise, the expression was simplified through several steps that highlighted the use of properties like the distributive law when combined with multiplication by fractions.

The expression $ \log \sqrt{10x} $ was manipulated into simpler terms by focusing on the fact that square roots can be expressed as fractional powers $ (\sqrt{a} = a^{1/2}) $.

This allows a logarithmic expression to be rewritten as $ \frac{1}{2}(\log 10 + \log x) $, facilitating direct comparison and simplification with $ \frac{1}{2} + \frac{1}{2} \log x $ in the solution.

Each step taken to simplify or transform a logarithmic expression brings us closer to discovering solutions or proving identities, as demonstrated in this exercise.

91影视

Explain why $$ \frac{1+\log x}{2}=\log \sqrt{10 x} $$ for every positive number \(x\)

Short Answer

Step by step solution

Rewrite the given equation

Simplify the square root using logarithmic properties

Distribute the fraction

Simplify the logarithmic expressions

Compare the expressions

Key Concepts

Logarithmic Properties

Equations

Logarithmic Expressions

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