Problem 10

Question

In $3-14,$ write each exponential equation in logarithmic form. $$ 4^{-2}=0.0625 $$

Step-by-Step Solution

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Answer

The logarithmic form is $ \\log_4(0.0625) = -2 $.

1Step 1: Understand the Exponential Form

The given equation is in the form of an exponential equation, which can generally be written as $ a^b = c $. In this case, $ a = 4 $, $ b = -2 $, and $ c = 0.0625 $.

2Step 2: Identify the Logarithmic Form

To convert the exponential form $ a^b = c $ to the logarithmic form, we use the formula: $ \log_a(c) = b $. This step translates the exponential equation into its equivalent logarithmic form.

3Step 3: Substitute Into the Logarithmic Form

Substitute the values from the exponential form into the logarithmic expression: $ \log_4(0.0625) = -2 $. This is the equation written in logarithmic form.

Key Concepts

Exponential EquationsLogarithmsConversion Between Exponential and Logarithmic Form

Exponential Equations

Exponential equations are foundational in mathematics and they appear quite regularly in various real-world applications. These equations have a distinct format, expressed in the form $ a^b = c $. In this setup, $ a $ is known as the base, $ b $ is the exponent, and $ c $ is the result. The exponential equation tells you that "base $ a $ raised to the power of $ b $ gives the result $ c $." For instance, in our example, $ 4^{-2} = 0.0625 $, the equation demonstrates that 4 raised to the power of -2 equals 0.0625.
Understanding how these components interact is crucial:

Base ($ a $): The primary number being multiplied.
Exponent ($ b $): Indicates how many times the base is multiplied by itself. A negative exponent, such as -2, implies dividing instead of multiplying.
Result ($ c $): The outcome of the exponential operation.

Exploring these relationships is key to mastering exponential equations. Students often engage with these equations in topics like exponential growth and decay, and they serve as a stepping stone to deeper mathematical topics.

Logarithms

Logarithms are the mathematical counterpart to exponential equations. Imagine logarithms as tools that help to "undo" exponentiation. The equation $ \log_a(c) = b $ expresses what exponent $ b $ yields $ c $, when $ a $ is the base. Logarithms offer a way to solve exponential equations for unknown exponents, converting multiplication into addition, which simplifies calculations.
The key elements involved in logarithms include:

Base ($ a $): As in exponential form, this is the primary number of interest in your logarithmic expression.
Argument ($ c $): The result of the exponentiation, which you aim to reach.
Exponent ($ b $): The outcome, showing "how many times" the base needs to multiply to achieve $ c $.

Mastering logarithms involves learning how to convert between logarithmic and exponential forms, leveraging logarithmic rules, and understanding their properties, such as the product and quotient rules. This knowledge is vital for calculus, complex algebra, and even in fields outside mathematics like acoustics and finance.

Conversion Between Exponential and Logarithmic Form

Converting between exponential and logarithmic forms is a critical skill in algebra that connects these two mathematical concepts. It pivots around the relationship that any exponential equation $ a^b = c $ can be expressed in logarithmic form as $ \log_a(c) = b $.
This conversion is straightforward when you follow these steps:

Identify the base $ a $ of the exponential expression. This remains the base in the logarithmic form.
Pinpoint the exponent $ b $, which becomes the output or result in the logarithmic form.
The result $ c $ of the exponential equation becomes the argument of the logarithm.

Learning to perform these conversions:

Simplifies solving equations where the unknown value is an exponent.
Enables understanding of logarithmic functions, paving the way for calculus and advanced mathematics.

This understanding fosters a deeper appreciation of the symbiotic relationship between exponential and logarithmic forms, where one is the inverse operation of the other. This conceptual link broadens our number sense and mathematical problem-solving toolkit.

Problem 10

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