Problem 101 - College Algebra Essentials

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{3} 7=\frac{1}{\log _{7} 3}\)

Step-by-Step Solution

Answer

The given equation \( \log _{3} 7 = \frac{1}{ \log _{7} 3} \) is true.

Step 1: Identify the statement

The equation that needs to be tested is \( \log _{3} 7 = \frac{1}{\log _{7} 3} \)

Step 2: Simplify the right side of the equation

The statement can be evaluated like this: If the property of logarithms \( \log _{a} b = \frac{1}{ \log _{b} a} \) is correctly understood and applied, everything will come down to understanding that \( \frac{1}{ \log _{7} 3} \) actually means the logarithm of 7 to the base 3, so we can rewrite it as \( \log _{3} 7 \).

Step 3: Compare the left side and the right side of the equation

On the left side we have \( \log _{3} 7 \) and on the right side we've now also got \( \log _{3} 7 \). They are exactly the same, therefore, the given equation is true.

Key Concepts

Understanding Logarithmic Equations

Logarithmic equations are algebraic expressions involving logarithms of variables or constants. To solve these equations, one must understand the properties and rules that govern logarithms. In our exercise, we deal with an equation that requires us to evaluate the equality of two expressions based on logarithmic functions. One crucial aspect in understanding logarithmic equations is recognizing various properties and knowing how to manipulate them.

For example, when we come to an equation like the one presented in the exercise, evaluate whether can be viewed as true by comparing left and right side. Through the lens of logarithm rules, if our equation follows these rules correctly, and the left and right sides of the equation contain the same value, then we confirm the equation as true.

When student encounters a logarithmic equation, it's essential to first identify the form of the logarithms involved and then apply the matching properties to simplify the expressions. Careful step-by-step simplification is the key to correctly solving the equation and validating its truthfulness.

Mastering Logarithm Rules

Logarithms can be intimidating at first glance, but their rules simplify complex relationships between numbers into manageable operations. Some fundamental logarithm rules include the product rule, quotient rule, power rule, and the change of base formula. Understanding and applying these rules can transform a seemingly complex logarithmic equation into a straightforward exercise.

Product Rule:

evokes the multiplication of two numbers within a logarithm.

Quotient Rule:

divides two logarithmic numbers.

Power Rule:

allows the exponent in a logarithm to be brought in front of the log as a multiplier, which can significantly simplify calculations. By mastering these rules, one can confidently approach and solve logarithmic equations without the trepidation often associated with this topic.

Change of Base Formula

The change of base formula is an indispensable tool in logarithms when we need to evaluate logs with bases other than what is easily computable or standard, like base 10 or base e (ln). The formula is expressed as can be used to rewrite logarithms in terms of bases that are more amenable to computation or comparison.

In our original exercise, the change of base property appears inherently when asserting that This property exhibits the symmetrical nature of logarithms concerning their bases and numbers and is central to a variety of logarithmic manipulations in both theoretical and practical applications of mathematics.

By using the change of base formula, students can solve logarithmic equations involving any base by converting them into a base that is more straightforward to work with, which aligns with our goal of demystifying the concept of logarithms and rendering them approachable to learners at all levels.