Problem 5 - Algebra 2

Question

Write each equation in exponential form. \(\log _{36} 6=\frac{1}{2}\)

Step-by-Step Solution

Answer

The equation in exponential form is \( 36^{\frac{1}{2}} = 6 \).

Step 1: Identify the Logarithmic Equation

You are given the logarithmic equation \( \log_{36} 6 = \frac{1}{2} \). This equation states that \( \frac{1}{2} \) is the power to which the base 36 must be raised to obtain the number 6.

Step 2: Convert Logarithmic to Exponential Form

To convert from a logarithmic equation \( \log_b a = c \) to an exponential form, use the identity \( b^c = a \). For the given equation, \( b = 36 \), \( c = \frac{1}{2} \), and \( a = 6 \).

Step 3: Write the Exponential Equation

Using the conversion identity, substitute the values into the equation \( b^c = a \). Thus, \( 36^{\frac{1}{2}} = 6 \). This is the exponential form of the given logarithmic equation.

Key Concepts

Logarithmic and Exponential Relationships

Logarithms and exponents are closely related mathematical concepts that often go hand-in-hand. Understanding their relationship is important because it lays the foundation for solving complex mathematical problems.
A logarithm is essentially the inverse operation of an exponent. When you see a logarithmic expression such as \( \log_b a = c \), it means you're asking "To what power do we need to raise \( b \) to get \( a \)?"
In contrast, an exponential expression, like \( b^c = a \), directly involves raising \( b \) to the power of \( c \) to achieve \( a \). This connection between logarithms and exponents allows for a two-way street in mathematics where you can convert equations from one form to the other.

Converting Logarithmic to Exponential

Converting between logarithmic and exponential forms is a crucial skill in algebra and precalculus. It involves using the identity \( \log_b a = c \) and corresponding it to the exponential form \( b^c = a \). Let's break it down:

• Identify the base \( b \), the argument \( a \), and the exponent \( c \) in the logarithmic form.
• Switch these components to fit into the exponential equation \( b^c = a \).

For example, with the logarithmic equation \( \log_{36} 6 = \frac{1}{2} \):
- The base \( b \) is 36.
- The result of the logarithmic operation (\( a \)) is 6.
- The exponent (\( c \)) is \( \frac{1}{2} \).
Thus, you convert the logarithmic equation into its exponential form: \( 36^{\frac{1}{2}} = 6 \). This shows that \( 36 \) raised to the power of \( \frac{1}{2} \) equals 6.

Logarithmic Equations

Solving logarithmic equations involves finding the value of the variable that makes the equation true. Logarithmic equations can often be transformed into exponential equations, which are sometimes easier to solve.
Here are some key tips for solving logarithmic equations:

• Re-arrange the equation if necessary to isolate the logarithm.
• If the logarithm is isolated, convert the logarithmic equation to its exponential form to simplify the equation.
• Use basic algebra to solve the resulting equation.

In our exercise, the logarithmic equation \( \log_{36} 6 = \frac{1}{2} \) was converted to its exponential counterpart \( 36^{\frac{1}{2}} = 6 \), which directly gives the answer without needing further calculations. Mastering these conversions can make solving logarithmic equations more intuitive.