Problem 816 Solve the equation \(2 \log \mat... [FREE SOLUTION]

Chapter 26: Problem 816

Solve the equation \(2 \log \mathrm{x}-\log 10 \mathrm{x}=0\).

Short Answer

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The short answer to the problem \(2 \log x - \log 10x = 0\) is: To solve for x, rewrite the equation and apply logarithm properties to get \(\log\frac{x^2}{10x} = 0\). Remove the logarithm to obtain \(\frac{x^2}{10x} = 1\). After solving for x, you get that \(x = 10\).

Step by step solution

Rewrite the given equation

Let's rewrite the given equation using a single logarithm on both sides of the equation: \(2 \log x - \log 10x = 0\)

Apply the logarithm properties

Using the logarithm properties, we can simplify the equation: - Logarithm Product rule: \(log_{a} (xy) = log_{a} x + log_{a} y\) - Logarithm Quotient rule: \(log_{a}(\frac{x}{y})=log_{a}(x)-log_{a}(y)\). - Logarithm power rule: \(log_{a}(x^y)=y路log_{a}(x)\) In our case, the equation can be simplified as follows: \(2 \log x - \log (10x) = \log(x^2) - \log (10x)\) Now, we can use the log quotient rule: \(\log\frac{x^2}{10x} = 0\)

Remove the logarithm

Since we have a logarithm equal to 0, we can remove the logarithm by applying the inverse operation, which is the exponential function: \(\frac{x^2}{10x} = 10^0\) As we know, any non-zero number to the power of 0 is equal to 1. Thus, \(\frac{x^2}{10x} = 1\)

Solve for x

To solve for x, we will first multiply both sides of the equation by 10x: \(x^2 = 10x\) Now, we move all the terms to one side of the equation: \(x^2 - 10x = 0\) Next, we can factor out x: \(x(x - 10) = 0\) Finally, we solve for x by setting each factor to zero: \(x = 0\) or \(x - 10 = 0\) However, since we cannot have a logarithm of 0, we discard \(x = 0\) as a solution. Therefore, the solution to the given equation is: \(x = 10\)

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Key Concepts

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

Logarithm Properties

Understanding logarithm properties is essential when solving logarithmic equations. Logarithms, simply put, are another way to express exponentiation. For instance, the logarithm of a number with a given base answers the question: 'To what power should the base be raised to obtain that number?'

There are several properties of logarithms that make calculations easier. These properties, including the product, quotient, and power rules, allow us to manipulate and combine logarithms in various ways. These manipulations often lead to simplifying the original equation into a more solvable form.

Logarithm Product Rule

The logarithm product rule is a property that helps combine two logarithms with the same base connected by multiplication into a single logarithm. In more technical terms, it is expressed as: \[\log_b(xy) = \log_b(x) + \log_b(y)\].

This rule is used when you want to combine logarithms of two multiplied numbers. By understanding and applying this rule, you can significantly simplify a problem before solving it.

Logarithm Quotient Rule

The logarithm quotient rule is similar to the product rule but for division. It provides a method for combining logarithms wherein one number is divided by another. The rule is stated as: \[\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\].

Using this property, we can simplify terms involving the logarithm of a quotient by expressing it as the difference of two logarithms, which often paves the way to an easier solution.

Logarithm Power Rule

The logarithm power rule is vital for handling exponents within a logarithm. It states that the logarithm of a number raised to a power is the same as the power multiplied by the logarithm of the number: \[\log_b(x^y) = y \cdot \log_b(x)\].

Essential for solving more complex logarithmic equations, this rule also helps to demystify seemingly intricate logarithm expressions by bringing the exponent down to a more manageable position. It plays a crucial role in breaking down the original equations into simpler parts.

Exponential Functions

Exponential functions are the inverse of logarithms and are used to rewrite logarithmic equations into a different form that can be solved more straightforwardly. The function is typically written as \[b^y = x\], which reads as 'b raised to the power of y equals x.'

When solving logarithmic equations, we sometimes set the equation equal to zero and then use the inverse property of logarithms to rewrite it in exponential form for easier solution. This approach allows us to determine the unknown variable without the logarithm involved, as seen in the step-by-step solution of the provided exercise.

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91影视

Solve the equation \(2 \log \mathrm{x}-\log 10 \mathrm{x}=0\).

Short Answer

Step by step solution

Rewrite the given equation

Apply the logarithm properties

Remove the logarithm

Solve for x

Key Concepts

Logarithm Properties

Logarithm Product Rule

Logarithm Quotient Rule

Logarithm Power Rule

Exponential Functions

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