Question

Rewrite the equation in logarithmic form. (See Example 2.) \(49^{1 / 2}=7\)

Short answer

The logarithmic form of the given expression \(49^{1 / 2} = 7\) is \(\log_{49}{7} = 1/2\).

Step-by-step solution

Identify the base, exponent, and result in the given expression

In the given expression \(49^{1 / 2} = 7\), '49' is the base, '1/2' is the exponent, and '7' is the result of this operation.

Apply the rule to convert from exponential to logarithmic form

By applying the rule \(\log_b{x} = y\), where 'b' is the base, 'x' is the result and 'y' is the exponent, we rewrite the expression \(49^{1 / 2} = 7\) in logarithmic form as \(\log_{49}{7} = 1/2\).

Key concepts

Exponential to Logarithmic Conversion

When you come across an equation like \(49^{1/2} = 7\), you can express it in logarithmic form to better understand the relationship between the base, exponent, and result. The key to conversion is using the fundamental relationship:

• If \(b^y = x\), then in logarithmic form, it is \(\log_b{x} = y\).

This relation shows that the logarithm \(\log_b{x}\) tells us what power \(b\) was raised to in order to get \(x\). In our example, the base \(49\) raised to the power of \(1/2\) results in \(7\). Hence, writing this in log form becomes \(\log_{49}{7} = 1/2\) which clearly displays the exponent.

Base, Exponent, and Result Identification

Identifying the base, exponent, and result in exponential expressions is crucial in mathematics, especially when converting to logarithmic form. Let's look at the expression \(49^{1/2}=7\).

• Base: The base is the number that is repeatedly multiplied. Here, it is \(49\).
• Exponent: The exponent tells you how many times to use the base in a multiplication. In this case, it is \(1/2\), which implies a square root.
• Result: The result is what you get after applying the exponent to the base, which here is \(7\).

Understanding these components helps in simplifying and manipulating mathematical equations, enabling easier conversions between forms.

Logarithmic Functions

Logarithmic functions serve as the inverse operations to exponential functions. They answer the question: "To what power must a certain base be raised to obtain a particular number?" Suppose we're working with \(\log_{49}{7} = 1/2\).

• Inverse Function: If an exponential expression is \(b^y = x\), the logarithmic form \(\log_b{x} = y\) reverses this relationship.
• Interpretation: In our example, logarithms help us express the fact that \(49\) raised to the power of \(1/2\) equals \(7\).
• Use Cases: Logarithms are extensively used in sciences, engineering, and mathematics to deal with exponential growth or decay, among other applications.

Hence, understanding logarithmic functions deepens our comprehension of their exponential counterparts and broadens our problem-solving tools.

Laws of Exponents

Understanding the laws of exponents is essential for manipulating and simplifying expressions, particularly when switching between exponential and logarithmic forms. For our expression \(49^{1/2}=7\), these laws are demonstrated as follows:

• Rule of Exponents: Raising a base to an exponent \(b^y\) gives us the result \(x\).
• Fractional Exponents: An exponent of \(1/2\) implies a square root, which transforms \(49^{1/2}\) into the square root of \(49\), resulting in \(7\).
• Simplification and Conversion: Knowing these rules assists in simplifying complex expressions, making them more manageable, and facilitates easy conversion to logarithmic form.

By grasping these principles, you can approach various mathematical problems with confidence and ease, improving both your computational and conceptual skills.