Problem 16 - Contemporary Precalculus

Question

Translate the given exponential statement into an equivalent logarithmic statement. $$10^{3}=1000$$

Step-by-Step Solution

Answer

Question: Convert the given exponential statement into an equivalent logarithmic statement: $$10^{3}=1000$$ Answer: The equivalent logarithmic statement for the given exponential statement is $$\log(1000)=3$$.

Step 1: Identify the base, exponent, and the result

In the given exponential statement $$10^{3}=1000$$, the base is 10, the exponent is 3, and the result is 1000.

Step 2: Write the logarithmic statement using the base, exponent, and result

We will now write the logarithmic statement using the base (a = 10), exponent (x = 3), and result (b = 1000). Recall the logarithmic function is written as $$\log_ab=x$$. Plugging in the values, we get: $$\log_{10}(1000)=3$$

Step 3: Simplifying the logarithmic statement

The logarithmic statement can be further simplified as: $$\log_{10}(1000)=3$$ is equivalent to $$\log(1000)=3$$, as it is understood that the base is 10 when the base is not mentioned explicitly in a logarithmic function. So, the given exponential statement, $$10^3=1000$$, can be converted into an equivalent logarithmic statement, which is $$\log(1000)=3$$.

Key Concepts

Exponents

Exponents are a shorthand notation used in mathematics to denote repeated multiplication of a number by itself. When we write \(10^3\), it means we multiply 10 by itself three times: \(10 \times 10 \times 10 = 1000\). Here, 10 is called the "base," and 3 is the "exponent" or the "power." Understanding exponents is crucial for simplifying expressions and solving equations, especially in higher level math and sciences.

There are a few key points about exponents that are important to know:

• Zero Exponent Rule: Any nonzero number raised to the power of zero equals one, i.e., \(a^0 = 1\).
• Negative Exponent Rule: A negative exponent represents the reciprocal of the number with a positive exponent, such as \(a^{-n} = \frac{1}{a^n}\).
• Multiplying Powers of the Same Base: When multiplying like bases, you add the exponents, i.e., \(a^m \times a^n = a^{m+n}\).
• Dividing Powers of the Same Base: When dividing like bases, you subtract the exponents, i.e., \(\frac{a^m}{a^n} = a^{m-n}\).

By mastering exponents, you create a strong foundation for dealing with more complex mathematical concepts.

Logarithmic Functions

Logarithmic functions serve as the inverse operations of exponential functions. In simple terms, while an exponent tells you the power a base needs to be raised to, a logarithm gives you the exponent itself. For instance, if you know that \(10^3 = 1000\), then \(\log_{10}(1000) = 3\) determines the exponent needed to raise 10 to achieve 1000.

Let's explore a few essential properties of logarithms:

• Logarithmic Identity: \(\log_a(a) = 1\).
• Product Rule: \(\log_a(xy) = \log_a(x) + \log_a(y)\).
• Quotient Rule: \(\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\).
• Power Rule: \(\log_a(x^n) = n \cdot \log_a(x)\).
• Change of Base Formula: \(\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\), which can be useful for calculations involving different bases.

Understanding these properties allows you to manipulate and solve logarithmic equations effectively. Logarithms are not only theoretical; they have vast applications, including measuring acidity (pH), decibels in sound, and in financial calculations.

Exponential Equations

Exponential equations are equations wherein variables appear as exponents. A typical example of an exponential equation is \(2^x = 16\). Solving such equations involves using both exponent rules and logarithms.

To solve exponential equations, you generally follow these methods:

• Matching Bases: If possible, express both sides of the equation with the same base. For instance, \(2^x = 16\) can be rewritten as \(2^x = 2^4\), allowing you to equate exponents: \(x = 4\).
• Using Logarithms: When matching bases isn't feasible, logarithms become a powerful tool. By taking the logarithm of both sides, you change the exponential equation into a form that can be solved using algebraic methods. For example, solve \(e^x = 5\) by using natural logarithms: \(\ln(e^x) = \ln(5)\), which simplifies to \(x = \ln(5)\).

These techniques provide a systematic approach to unravel exponential equations, which are prevalent in scientific models such as population growth, radioactive decay, and interest calculations in finance. By mastering these, you prepare yourself for engaging with dynamic, real-world phenomena.