Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation for \(x.\) $$\log _{7} x=\log _{7} 9$$

Step-by-Step Solution

Answer

The value of \(x\) that satisfies the given equation is \(x = 9\).

Step 1: Identify like terms

In this equation, \(\log _{7} x=\log _{7} 9\), \(\log _{7} x\) and \(\log _{7} 9\) are the like terms. They can directly be equated to each other since they have the same log base.

Step 2: Set the arguments of the logs equal

Next, the arguments of the logs are set equal and solved for \(x\). Thus, the equation becomes \(x = 9\).

Step 3: Verify

In this case, there's no need for verification, because the value of \(x\) makes the original equation true, and doesn't cause the logarithms to have negative or zero arguments.

Key Concepts

Logarithmic Properties

Logarithms are mathematical expressions that answer the question: 'To what power must a given base number be raised, to produce a certain number?' They come with a set of properties that make it easier to solve logarithmic equations.

Some essential logarithmic properties include:

• The product property: \( \log_b(x) + \log_b(y) = \log_b(xy) \).
• The quotient property: \( \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) \).
• The power property: \( n\cdot\log_b(x) = \log_b(x^n) \) where \(n\) is a real number.
• Change of base formula: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), for a new base \(k\).

Understanding these properties allows us to manipulate and solve logarithmic equations by transforming them into more workable forms. In the given exercise, the property that if \( \log_b(x) = \log_b(y) \), then \( x = y \) is used. This is because logarithms with the same base can be set equal to each other and solved directly, which makes finding the value of \(x\) simpler in this case.

Exponential Functions

Exponential functions are closely related to logarithms since they are basically the 'inverse' – in simple terms, the opposite. An exponential function in math is a function that rapidly increases, or 'explodes' in value, as the variable increases. It can be written in the general form: \( y = b^x \), where \( b \) is a positive real number, and \( x \) is the exponent.

Key features of exponential functions include:

• They never result in a negative number or zero; \( b^x > 0 \) for all real \(x\).
• The rate of growth or decay of the function is proportional to its current value.
• Exponential functions have a constant base and a variable exponent.

When solving logarithmic equations, knowing the inverse relationship between logarithms and exponentials is crucial because this allows us to convert from logarithmic to exponential form to isolate the variable and find solutions.

Algebraic Solutions

Algebra is the field of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. The process of finding the values that satisfy an equation is called solving the equation. Algebraic solutions often require a step-by-step methodology that leverages the properties of arithmetic operations and equalities.

When dealing with logarithmic equations, algebraic solutions involve making use of logarithmic properties to isolate the variable of interest. The basic steps to solve algebraic equations include:

• Combining like terms.
• Using the distributive property to remove parentheses.
• Isolating the variable on one side of the equation.
• Applying inverse operations to find the solution set.

In our exercise using algebraic methods, \(x\) was easily isolated by recognizing that both sides of the equation had the same logarithmic base, allowing for a direct comparison of their arguments. This direct comparison is a simplified version of an algebraic solution that we often use when the terms of an equation are similar.