Problem 13 - College Algebra

Question

For the following exercises, rewrite each equation in exponential form. $$\log _{13}(142)=a$$

Step-by-Step Solution

Answer

The equation in exponential form is \( 13^a = 142 \).

Step 1: Identify the Components of the Logarithm

The given logarithmic equation is \( \log_{13}(142) = a \). This equation is read as "logarithm base 13 of 142 equals \( a \)." Here, 13 is the base of the logarithm, 142 is the argument, and \( a \) is the result.

Step 2: Convert to Exponential Form

To convert a logarithmic equation into exponential form, recall that \( \log_{b}(x) = y \) can be rewritten as \( b^y = x \). Using this, convert the given equation: \( \log_{13}(142) = a \) becomes \( 13^a = 142 \).

Step 3: Verify the Exponential Equation

Check that the conversion has been done correctly by matching the base and arguments: make sure that the base (13) and exponent (\( a \)) match with the result (142), confirming that \( 13^a = 142 \) is correct.

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve a logarithm with a variable. In the expression \( \log_b(x) = y \), the "log" stands for logarithm, "b" is the base of the logarithm, "x" is the argument, and "y" is the result. They are a clever way to express exponential relationships. Understanding the interchangeable nature of logarithmic and exponential forms is crucial:

• Logarithmic form: \( \log_b(x) = y \)
• Exponential form: \( b^y = x \)

In our given problem, we have \( \log_{13}(142) = a \). This equation means "how many times do we multiply 13 by itself to get 142?". Converting logarithms into an exponential form is often necessary for solving and understanding the dynamics of how the numbers interact.

Base and Exponent

In both logarithmic and exponential equations, the base and the exponent play critical roles. With logarithms, the base \( b \) dictates the rate of growth or decay in exponential equations. A higher base results in faster growth. In our example, the base is 13.The exponent is the number "a" we are solving for, as it tells us how many times the base is used as a factor to reach the argument. Rewritten in exponential form (as derived), the equation becomes \( 13^a = 142 \). This means that 13 multiplied by itself "a" times gives us 142.Understanding the base and exponent can help to deduce and transform equations:

• The base indicates repeated multiplication.
• The exponent indicates the power to which the base is raised.
• The argument is the resulting value of such repeated multiplication.

Mathematical Argument

The mathematical argument in logarithmic and exponential equations is the quantity associated with the operation. It is the number that the base, when raised to the exponent, equals. In our problem, the argument is 142.The argument is what we seek to reach using our base and exponent in the exponential form \( b^y = x \). Understanding its role helps in verifying and solving these equations effectively.In practical terms, the argument in \( \log_{13}(142) = a \) is an expression of the ultimate goal or result in multiplication terms. When we shift from logarithmic to exponential form, it helps us easily verify correctness by matching it with the value derived from equating the base and exponent.When working with equations, always:

• Identify the argument.
• Ensure the argument matches the result of your exponential expression.
• Re-evaluate the equation whenever there seems to be a discrepancy.