Question

Rewrite the equation in logarithmic form. (See Example 2.) \(125^{2 / 3}=25\)

Short answer

\(\log_{125} 25=2 / 3\)

Step-by-step solution

Identify the parts

In the written equation, identify the base (b), the exponent (n), and the value to which it equals (m). In this case, \(b=125\), \(n=2 / 3\), and \(m=25\).

Apply the logarithmic conversion

Replace the parts of the exponential form to the relevant parts in logarithmic form. This swaps from the form \(b^n=m\) to the form \(\log_b m=n\). As per this format, convert the given equation to: \(\log_{125} 25=2 / 3\).

Key concepts

Exponential to Logarithmic Conversion

Converting an equation from its exponential form into a logarithmic form is a valuable skill. It allows you to express relationships in a different but equivalent manner. An exponential equation has the structure \(b^n = m\), where \(b\) is the base, \(n\) is the exponent, and \(m\) is the result.

To convert it to a logarithmic form, you rearrange the parts of the equation to highlight a different perspective: \(\log_b m = n\). This conversion shows that the exponent \(n\) is what you need to raise the base \(b\) to get \(m\).

In our example, \(125^{2/3} = 25\) can be converted by identifying what each part represents and swapping them into the logarithmic form \(\log_{125} 25 = 2/3\). This new form focuses on determining the power 2/3 that transforms \(125\) into \(25\).

• Exponential form: Shows the straightforward calculation \(b^n = m\).
• Logarithmic form: Emphasizes the power needed for base \(b\) to achieve \(m\).

Base and Exponent Identification

Before converting between exponential and logarithmic forms, it's essential to correctly identify the base, exponent, and result in the equation. This understanding forms the foundation for accurate conversion.

For the equation \(125^{2/3} = 25\):
- The base \(b\) is the number being raised to a power, which in this case is \(125\).
- The exponent \(n\) is the power applied to the base, here expressed as \(2/3\).
- The result \(m\) is what the expression evaluates to, which is \(25\) in this example.

Correct identification ensures you can accurately apply the conversion process. Recognizing the base and exponent is crucial not just in rewriting equations but also in understanding how certain mathematical operations work.

• Base: The number we multiply by itself "n" times.
• Exponent: The power to which the base is raised.
• Result: The outcome of the operation.

Conversion of Equations

Conversion between different forms of equations enhances mathematical versatility and problem-solving ability. Shifting from exponential form to logarithmic form, or vice versa, depends heavily on the proper recognition and comprehension of each part of the expression.

The steps to convert \(125^{2/3} = 25\) into its logarithmic counterpart \(\log_{125} 25 = 2/3\) were straightforward because we clearly identified all components. This conversion brings forth new insights and solutions, such as:

• Simplifying complex numbers and expressions.
• Solving for unknowns that are exponents or bases.
• Understanding growth and decay problems in calculus.
• Applying logarithmic scales in scientific equations.

Understanding each equation form and how they relate to each other expands analytical skills, allowing you to tackle a variety of mathematical challenges with confidence.