Problem 33 - Algebra 2

Question

Write each equation in logarithmic form. \(100^{\frac{1}{2}}=10\)

Step-by-Step Solution

Answer

\(\log_{100}{10} = \frac{1}{2}\)

Step 1: Understand the Equation

We have the equation \(100^{\frac{1}{2}}=10\). This equation is in exponential form and needs to be converted to logarithmic form.

Step 2: Recall the Logarithmic Form

The logarithmic form of an equation is \(\log_b{y} = x\), where \(b^x = y\). This means \(b\) is the base, \(x\) is the exponent, and \(y\) is the result.

Step 3: Identify the Base, Exponent, and Result

In the equation \(100^{\frac{1}{2}}=10\), identify the base as 100, the exponent as \(\frac{1}{2}\), and the result as 10.

Step 4: Plug Values into Logarithmic Form

Using the logarithmic form \(\log_b{y} = x\), substitute the identified values: base \(b = 100\), result \(y = 10\), and exponent \(x = \frac{1}{2}\).

Step 5: Write the Final Logarithmic Equation

Substituting the values into the logarithmic form gives \(\log_{100}{10} = \frac{1}{2}\). This is the logarithmic form of the original equation.

Key Concepts

Exponential Equations

An exponential equation is one where a variable appears in the exponent. These equations express how a base number is raised to a power or exponent to produce a result. They look like this: \(b^x = y\), where \(b\) is the base, \(x\) is the exponent, and \(y\) is the result. In the exercise example, the exponential equation is \(100^{\frac{1}{2}} = 10\). Here, 100 is the base, \(\frac{1}{2}\) is the exponent, and 10 is the result. Exponential equations are powerful in describing real-world phenomena like population growth, radioactive decay, and interest rates. Recognizing the base, exponent, and result is vital when solving these equations. It helps us convert them into a form that we can solve or manipulate more easily.

Logarithms

Logarithms are the inverse operations of exponentials. They are specifically used to unravel exponential equations by transforming them into a format where the exponent comes in handy directly. If you have an equation \(b^x = y\), the logarithm helps to solve for \(x\) by expressing it as \(\log_b{y} = x\). Here, \(b\) is the base of the logarithm, \(y\) is the result, and \(x\) is the exponent you are solving for.Using logarithms can make complex calculations more straightforward. For instance, solving equations where exponents involve unknowns becomes manageable by transforming into a logarithmic format. They also have properties that make computation simpler, such as the product, quotient, and power rules for logarithms:

• 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: \(\log_b(M^k) = k\log_b{M}\)

Understanding these fundamentals can immensely aid in dealing with exponential equations.

Converting Exponential to Logarithmic Form

Converting an exponential equation into its logarithmic form is an essential skill, as it transforms complex power relations into simpler linear equations. To convert, use the basic relationship: if you have \(b^x = y\), the equivalent logarithmic form is \(\log_b{y} = x\).Let's take the exercise's exponential equation \(100^{\frac{1}{2}} = 10\) as an example.

• Identify the base (\(b = 100\)), the exponent (\(x = \frac{1}{2}\)), and the result (\(y = 10\)).
• Substitute these values into the logarithmic relationship \(\log_b{y} = x\).
• We put these together to get \(\log_{100}{10} = \frac{1}{2}\).

Thus, the conversion translates the exponential nature of the original equation into a clear logarithmic format. This form is often more convenient for understanding and solving practical problems. Connecting and converting between these two forms allows for easier manipulation and solution finding, making logarithmic forms invaluable in mathematics.