Problem 21 - Precalculus Mathematics for Calculus

Question

Evaluate the expression. (a) \(\log _{6} 36\) (b) \(\log _{9} 81\) (c) \(\log _{7} 7^{10}\)

Step-by-Step Solution

Answer

(a) 2, (b) 2, (c) 10

Step 1: Identify the First Expression

The first expression to evaluate is \( \log _{6} 36 \). This asks what power we raise 6 to, to get 36.

Step 2: Express as an Exponential Equation

We translate the logarithmic expression \( \log _{6} 36 = x \) into the exponential form \( 6^x = 36 \). We need to find the value of \( x \).

Step 3: Solve the Exponential Equation

Recognize that \( 36 \) can be written as \( 6^2 \). Therefore, \( 6^x = 36 \) implies \( 6^x = 6^2 \), so \( x = 2 \). Thus, \( \log _{6} 36 = 2 \).

Step 4: Identify the Second Expression

The next expression to evaluate is \( \log _{9} 81 \). This asks what power we raise 9 to, to get 81.

Step 5: Express as an Exponential Equation

Translate \( \log _{9} 81 = y \) into \( 9^y = 81 \). We need to find \( y \).

Step 6: Solve the Exponential Equation

Recognize that \( 81 \) can be expressed as \( 9^2 \). So \( 9^y = 9^2 \) implies \( y = 2 \). Thus, \( \log _{9} 81 = 2 \).

Step 7: Identify the Third Expression

The final expression to evaluate is \( \log _{7} 7^{10} \). This asks what power we raise 7 to, to get \( 7^{10} \).

Step 8: Use the Property of Logarithms

Utilize the logarithmic property \( \log _{b} b^n = n \). Here, \( b = 7 \) and \( n = 10 \), so \( \log _{7} 7^{10} = 10 \).

Key Concepts

Exponential Equations

Exponential equations are equations where the variable is in the exponent, such as \( b^x = a \). To solve these, we often need to relate the bases of the exponents. For example, if we have \( 6^x = 36 \), we can express 36 as \( 6^2 \). Now, comparing the exponents directly gives us \( x = 2 \).

Solving exponential equations typically involves the following steps:

• Identify the equation: Recognize that the equation is in exponential form.
• Simplify to a common base if possible: Express numbers in the equation as powers of the same base.
• Set the exponents equal to each other: Once the bases are equal, equate the exponents.
• Solve for the variable: Solve the resulting simple equation to find the value of the variable.

With practice, understanding these steps becomes more straightforward, simplifying the process of solving exponential equations.

Logarithmic Expressions

Logarithmic expressions are used to determine the power to which a number (the base) must be raised to yield a given number. For instance, \( \log_{6} 36 \) asks "to what power must 6 be raised to result in 36?" The answer is \( 2 \), since \( 6^2 = 36 \).

Logarithms are often used because they transform multiplicative relationships into additive ones. This makes calculations easier and more intuitive.

• Recognize the structure: The expression \( \log_{b} a \) implies solving for \( x \) in \( b^x = a \).
• Apply logarithmic identities: Use logarithmic identities, such as \( \log_{b} (b^x) = x \), to simplify expressions.

Understanding logarithms allows us to easily switch back and forth between exponential and logarithmic forms.

Properties of Logarithms

Properties of logarithms are rules that simplify expressions and solve logarithmic equations. These properties include power, product, and quotient rules.

• The power rule: \( \log_{b} (a^n) = n \log_{b} a \). Simplifies the handling of powers in logarithms.
• The product rule: \( \log_{b} (mn) = \log_{b} m + \log_{b} n \). Adds logs of multiplied numbers together.
• The quotient rule: \( \log_{b} \left(\frac{m}{n}\right) = \log_{b} m - \log_{b} n \). Logs of divided numbers become a subtraction.
• The change of base formula: Useful when converting from one logarithmic base to another, such as \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \).

These rules not only help simplify complex logarithmic expressions but also assist in solving equations much quicker.

Solving Logarithmic Equations

Solving logarithmic equations often involves using the properties of logarithms and understanding their relationship to exponentials. An example equation is \( \log_{7} 49 = x \), which translates to the exponential form \( 7^x = 49 \). In this case, \( x = 2 \) because \( 49 = 7^2 \).

To solve logarithmic equations, consider these steps:

• Identify the logarithmic form and express it as an exponential equation.
• Simplify the equation using logarithmic properties if needed.
• Use the relationship between exponentials and logarithms to isolate the variable.
• Calculate the variable's value.

Having a solid grasp of both logarithmic and exponential forms enables one to solve logarithmic equations efficiently.