Solving Logarithm Equation for x

We are given the equation:

\(\text{log}_{10} x + \text{log}_{10} x^2 = 2 \text{log}_{10} x + 1\)

Our goal is to find the value of \(x\) that satisfies this logarithm equation.

Applying Logarithm Properties

To solve this equation, we need to use the properties of logarithms. The relevant properties here are:

• Product Rule: \(\text{log}_b (MN) = \text{log}_b M + \text{log}_b N\)
• Power Rule: \(\text{log}_b (M^p) = p \text{log}_b M\)

Using the Power Rule, we can rewrite the term \(\text{log}_{10} x^2\) on the left side of the equation:

\(\text{log}_{10} x^2 = 2 \text{log}_{10} x\)

Now, substitute this back into the original equation:

\(\text{log}_{10} x + (2 \text{log}_{10} x) = 2 \text{log}_{10} x + 1\)

Simplifying and Solving the Equation

Combine the terms on the left side of the equation:

\((1 \text{log}_{10} x) + (2 \text{log}_{10} x) = 3 \text{log}_{10} x\)

So the equation becomes:

\(3 \text{log}_{10} x = 2 \text{log}_{10} x + 1\)

Now, we need to isolate the term \(\text{log}_{10} x\). Subtract \(2 \text{log}_{10} x\) from both sides of the equation:

\(3 \text{log}_{10} x - 2 \text{log}_{10} x = 1\)

This simplifies to:

\(\text{log}_{10} x = 1\)

Converting to Exponential Form

The final step is to convert this logarithmic equation into its equivalent exponential form. The definition of a logarithm states that if \(\text{log}_b M = c\), then \(b^c = M\).

In our equation, \(\text{log}_{10} x = 1\), the base is \(b=10\), the argument is \(M=x\), and the value is \(c=1\).

So, converting to exponential form, we get:

\(10^1 = x\)

Therefore, the value of \(x\) is:

\(x = 10\)

Checking the Solution

It's always a good practice to check if the value of \(x\) is valid for the original logarithm equation. The arguments of logarithms must be positive. For \(x=10\), \(x\) is positive, and \(x^2 = 10^2 = 100\) is also positive. So, \(x=10\) is a valid solution.

Let's substitute \(x=10\) into the original equation:

Left side: \(\text{log}_{10} 10 + \text{log}_{10} 10^2 = \text{log}_{10} 10 + \text{log}_{10} 100\)

Using the fact that \(\text{log}_{10} 10 = 1\) and \(\text{log}_{10} 100 = 2\) (since \(10^2 = 100\)):

Left side = \(1 + 2 = 3\)

Right side: \(2 \text{log}_{10} x + 1 = 2 \text{log}_{10} 10 + 1\)

Right side = \(2(1) + 1 = 2 + 1 = 3\)

Since the left side equals the right side (\(3=3\)), the solution \(x=10\) is correct.

The value of \(x\) is 10.

Revision Table: Logarithm Equation Solution

Additional Information: Logarithm Concepts

Definition of Logarithm: The logarithm of a number \(M\) to a base \(b\) is the exponent \(c\) to which the base must be raised to produce \(M\). This is written as \(\text{log}_b M = c\), which is equivalent to \(b^c = M\). The base \(b\) must be a positive number other than 1, and the argument \(M\) must be a positive number.

Common Logarithm: When the base of the logarithm is 10, it is called the common logarithm. It is often written as \(\text{log} x\) (without the base subscript) or \(\text{log}_{10} x\).

Key Logarithm Properties Used:

• Product Rule: \(\text{log}_b (MN) = \text{log}_b M + \text{log}_b N\) (Used to understand the structure, though simplification directly used Power Rule).
• Power Rule: \(\text{log}_b (M^p) = p \text{log}_b M\) (Crucial for simplifying \(\text{log}_{10} x^2\)).
• Base Case: \(\text{log}_b b = 1\) (Used in checking the solution, \(\text{log}_{10} 10 = 1\)).