Problem 15 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$ 13^{2}=x $$

Step-by-Step Solution

Answer

The logarithmic form of the given equation is \(\log_{13}x=2\)

Step 1: Identify the base, exponent and the result

In the given exponential equation, \(13^{2}=x\), the base is 13, the exponent is 2 and the result is \(x\).

Step 2: Convert the exponential equation to logarithmic form

Therefore, the logarithmic form is given by \(\log_{13}x=2\) as per the definition of logarithms.

Key Concepts

Exponential Equations

Exponential equations are equations where a variable appears in the exponent. A classic example is the equation \(13^2 = x\). Here, the number 13 is the base, 2 is the exponent, and \(x\) is the result or power of the base raised to the exponent. Exponential equations are powerful mathematical expressions that describe growth and decay processes in fields like finance and physics.
When dealing with exponential equations, you also deal with specific rules of manipulation. This can involve the use of properties such as:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m\cdot n}\)
• Zero Power: Any base raised to the power of zero is 1, \(a^0 = 1\)

These rules help simplify the process of solving or transforming exponential equations. Recognizing these patterns will aid you significantly as you convert between exponential and logarithmic forms.

Logarithms

Logarithms are the inverse operations of exponentials, just like subtraction is the inverse of addition. Specifically, a logarithm answers the question, "To what exponent must the base be raised, to produce a given number?" For example, in the expression \( \log_{13} x = 2 \), you are asking, "What power should 13 be raised to, in order to get \(x\)?" In this case, the answer is 2.Logarithms have their properties and rules that can be utilized to simplify complex equations:

• Product Rule: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b (M^p) = p \cdot \log_b M \)

Understanding these rules of logarithms can optimize your problem-solving process, especially when dealing with exponential values in real-world situations.

Converting Between Exponential and Logarithmic Form

Converting between exponential and logarithmic forms is a fundamental skill in mathematics. It involves rewriting an equation to express it either in exponential or logarithmic terms. The exponential form \(a^b = c\) can be converted into its logarithmic equivalent as \(\log_a c = b\).Let's break down how to convert the example from the exercise:

• Identify the base of the exponential equation. In \(13^2 = x\), the base is 13.
• Recognize the exponent. The exponent here is 2.
• Identify the result or outcome of the exponential expression, which is \(x\).
• Apply the definition of logarithms to rewrite: \(\log_{13} x = 2\). This signifies that raising 13 to the power of 2 results in \(x\).

This conversion is particularly useful because logarithmic form can simplify solving for unknowns in equations, especially those involved with exponential growth or decay scenarios. Mastering this conversion will allow you to handle a wide range of mathematical problems more easily.