Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$n^{t}=10$$

Step-by-Step Solution

Answer

The logarithmic form of given expression \(n^{t}=10\) is \(\log _n 10 = t\).

Step 1: Identify the components of the exponential equation

In the exponential form \(n^{t}=10\), \(n\) is the base, \(t\) is the exponent, and \(10\) is the result of the base raised to the exponent-power.

Step 2: Recall the relationship between exponential and logarithmic form

The logarithmic form of an exponential equation \(b^c = a\) is \(\log _b a = c\). Here, \(b\) is the base, \(c\) is what the base is raised to and \(a\) is the result of the base raised to that power.

Step 3: Transform into logarithmic form

Applying the conversion rule to the given exponential equation \(n^{t}=10\), the logarithmic form will be \(\log _n 10 = t\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a number, called the base, is raised to the power of an exponent. These equations are used to describe phenomena such as exponential growth or decay, where something increases or decreases at a consistent rate over time. In the equation \(n^{t}=10\), \(n\) represents the base, \(t\) is the exponent or the power, and the result is the value obtained by raising the base to the power of the exponent.

Exponential equations are widely encountered in various fields

• In finance, for calculating compound interest.
• In biology, for modeling population growth.
• In physics, when dealing with radioactive decay.

Solving exponential equations often requires techniques such as isolating the variable, applying logarithms, or using exponent rules. For example, if the equation cannot be solved through simple algebraic manipulations, converting the equation into a logarithmic form might be necessary.

Understanding exponential equations and their transformations is essential in mastering algebra and aids in understanding real-world applications.

Logarithms

Logarithms are mathematical operations that are essentially the inverse of exponentiation. While an exponent determines how many times a number (the base) is multiplied by itself, a logarithm takes the result of this operation and tells you the original power to which the base was raised. If you have the equation \(b^c = a\), its logarithmic form is expressed as \(\log_b a = c\).

This transforms exponential difficulties into manageable algebraic challenges. Using logarithms:

• Simplifies solving exponential equations.
• Helps in calculating growth rates and decay rates over a continuous time interval.
• Facilitates data visualization in certain types of graphs, like logarithmic scales.

In our context, converting the exponential equation \(n^{t}=10\) into logarithmic form using the base \(n\), results in the expression \(\log_n 10 = t\). This tells us that the power \(t\) we need to raise \(n\) to, in order to get 10, is the solution.

By learning logarithms, you can gain powerful tools for solving complex equations and better understanding the relationships between numbers.

Mathematical Transformations

Mathematical transformations involve changing the form of an expression or equation without altering its fundamental relationships or characteristics. This often includes transformations such as translating exponential equations into logarithmic form. Each transformation brings its unique advantages in simplifying the problem solving process or providing insights into the relationship described by the equation.

Transformations can:

• Make complex equations more manageable.
• Allow for the solution of problems that are otherwise difficult to tackle.
• Reveal underlying relationships and properties.

In the example equation \(n^{t} = 10\), transforming it to the logarithmic form \(\log_n 10 = t\) is a perfect illustration of how such a change can make finding the value of \(t\) a straightforward task through the power of algebraic manipulation. Understanding the rules and strategies for different mathematical transformations is crucial for solving advanced mathematical problems and for developing a versatile approach to problem-solving in mathematics.