Problem 3 - Algebra for College Students

Question

Write each exponential statement in logarithmic form. For example, \(2^{5}=32\) becomes \(\log _{2} 32=\) 5 in logarithmic form. (c) \(\left(g^{-1} \circ f^{-1}\right)(x)\) \(\frac{x+7}{6}\) $$ 5^{3}=125 $$

Step-by-Step Solution

Answer

\(\log_{5}(125) = 3\)

Step 1: Identify the base, exponent, and result

In the exponential equation, identify the base (5), the exponent (3), and the result (125). This will guide the conversion into a logarithmic form.

Step 2: Rewrite in logarithmic form

The general form of a logarithmic equation is \ \log_{\text{base}}(\text{result}) = \text{exponent} \. For the given equation \(5^3 = 125\), express it as \(\log_{5}(125) = 3\). This means: 'the power you need to raise 5 to get 125 is 3.'

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a number, known as the base, is raised to a power, called the exponent. These equations are found in various fields including science, finance, and engineering. Understanding them can help solve many practical problems.

For example, in the equation \(5^3 = 125\), the base is 5 and the exponent is 3. The equation tells us that multiplying 5 by itself three times results in 125. Exponential equations help model growth and decay processes, like population growth, radioactive decay, and investment interests. They are vital because they allow for the expression of large values through repeated multiplication using a much simpler notation.

One of the first steps in dealing with exponential equations is correctly identifying the base and exponent, which are essential for converting into other forms like logarithms.

Base-Exponent-Result Identification

Identifying the base, exponent, and result is crucial when converting between exponential and logarithmic forms. Let's break down these three components using an equation like \(5^3 = 125\).

• Base: The base in this equation is 5. It is the number being repeatedly multiplied.
• Exponent: The exponent is 3. It indicates how many times the base is multiplied by itself.
• Result: The result is 125, which is the outcome of multiplying 5 by itself three times.

In any exponential equation, make sure you can clearly identify these three components. This breakdown forms the foundation needed to rewrite the equation in logarithmic form. Recognizing these components quickly becomes second nature with practice and helps streamline solving mathematical problems.

Logarithmic Equations

Logarithmic equations provide a way to reverse exponential operations. They are very helpful in solving exponential problems by converting them into a format that's easier to manipulate and solve where the unknown is an exponent.

To convert an exponential statement like \(5^3 = 125\) into a logarithmic form, you use the general form \(\log_{\text{base}}(\text{result}) = \text{exponent} \).

For our example, this means converting it into \(\log_{5}(125) = 3\). This can be interpreted as "5 raised to what power equals 125?" which clearly matches the exponent used in the exponential form.

Logarithms, therefore, simplify working with equations involving exponentials, allowing you to solve for unknown exponents more efficiently. They are especially useful when dealing with very large or small numbers, providing insight into growth rates, sound intensity, pH levels, and more. Understanding logarithmic equations is essential for navigating advanced mathematics and real-world applications.